= Solution
Throughout, <modules> are finite-dimensional over $k$, or finitely generated $\mathcal O$-free lattices when the coefficient ring is $\mathcal O$. The <p-modular system> has a complete <discrete valuation ring> $\mathcal O$, maximal ideal $(\pi)$, fraction <field> $K$ of characteristic zero and residue <field> $k$ of characteristic $p$.
A <block of a group algebra> is a nonzero, indecomposable two-sided ideal direct summand of $A=RG$. Equivalently it is $Ae$ for a primitive <central idempotent> $e$. To see the equivalence, suppose $A=B\oplus C$ as two-sided <ideals> and write $1=e+f$ with $e\in B$, $f\in C$. Products between $B$ and $C$ vanish because they lie in their zero intersection. Thus $e$ is the identity on $B$, zero on $C$, and is a <central idempotent>, with $B=Ae$. Conversely $A=Ae\oplus A(1-e)$. Splitting $Ae$ further is equivalent to splitting $e$ into orthogonal nonzero <central idempotents>; indecomposability is exactly primitivity. If $1=\sum e_i$ is the block decomposition, $M=\bigoplus e_iM$. A nonzero <indecomposable module> therefore belongs to exactly one <block of a group algebra>, namely the one with $e_iM=M$.
For <modular reduction of block idempotents>, the conjugacy class sums are an $R$-basis of $Z(RG)$. Hence $Z(\mathcal OG)/\pi Z(\mathcal OG)\cong Z(kG)$. In the complete commutative <algebra> $Z(\mathcal OG)$, <idempotent lifting> gives a unique lift of each <idempotent> of $Z(kG)$; uniqueness follows because commuting idempotents congruent modulo the <Jacobson radical> are equal. Orthogonal decompositions lift too, so primitivity is preserved. \b[Reduction gives a bijection between the integral and modular blocks.]
Here is a definition of <defect groups of a block> which also proves the required relative-projectivity assertion. Give $RG$ its conjugation action and, for $H\le G$, define the <relative trace> and its <transfer ideal of conjugation-fixed elements> by
$$
\operatorname{Tr}_H^G(a)=\sum_{g\in G/H}gag^{-1},\qquad a\in(RG)^H,\qquad I_H=\operatorname{Tr}_H^G((RG)^H)\subseteq Z(RG).
$$
A <defect group of a block> $RGe$ is a $p$-subgroup $D$ minimal with $e\in I_D$. Such a subgroup exists: for a <Sylow subgroup> $P$, $[G:P]$ is a unit in $R$ and $e=\operatorname{Tr}_P^G(e/[G:P])$. The <defect of a block> is $d$ where $|D|=p^d$.
For completeness, this definition is independent of a choice up to conjugacy. Multiplying two <relative traces> and grouping the pairs of cosets by <double cosets> gives
$$
I_D I_E\subseteq\sum_{g\in D\backslash G/E}I_{D\cap{}^gE}.
$$
This identity follows by writing each pair of cosets as a simultaneous $G$-orbit; its stabilizer is $D\cap{}^gE$, and its contribution is a trace from that stabilizer. If $D,E$ are both minimal for $e$, then $e=e^2$ belongs to the displayed sum. The commutative block center $eZ(RG)$ is a <local ring>: in the field case it is a commutative <Artinian ring> with no nontrivial <idempotents>; in the integral case its reduction has this property and completeness detects units. A sum of proper ideals cannot contain its identity. Consequently $e\in I_{D\cap{}^gE}$ for some $g$, forcing $D={}^gE$ by minimality. The same argument with an arbitrary $E$ shows that $e\in I_E$ precisely when $E$ contains a conjugate of $D$. This is the <trace criterion for defect groups>.
To relate this to the usual modular definition, the <Brauer morphism> deletes the coefficients outside the <centralizer>:
$$
\operatorname{Br}_P:(kG)^P\longrightarrow kC_G(P),\qquad\sum a_g g\longmapsto\sum_{g\in C_G(P)}a_g g.
$$
It is an <algebra homomorphism>: in the coefficient of an element centralizing $P$ in a product, $P$ acts on the contributing pairs; nonfixed orbits have size divisible by $p$, while fixed pairs have both entries in $C_G(P)$. Its kernel is the sum of proper-subgroup transfers, because the nonfixed conjugation orbit sums form a basis of that kernel. If $D$ is minimal for $e=\operatorname{Tr}_D^G(a)$ and $\operatorname{Br}_D(e)=0$, replace $a$ by $ea$. The kernel description gives $e\in\sum_{Q<D}I_Q$, contradicting the local-center argument. Also $\operatorname{Br}_P(I_D)=0$ unless $P$ is conjugate into $D$: group the cosets of $G/D$ into $P$-orbits, which have no fixed point otherwise. Thus the minimal transfer subgroups are exactly the maximal $p$-subgroups with nonzero <Brauer morphism> of $e$.
Now take $M=eM$ and write $e=\operatorname{Tr}_D^G(a)$. Multiplication by $a$ is a $D$-linear endomorphism $\alpha$ of $M$, and
$$
\operatorname{Tr}_D^G(\alpha)=\operatorname{id}_M.
$$
Here is the splitting behind the <D. Higman criterion>. The natural map $\varepsilon:RG\otimes_{RD}M\to M$, $g\otimes m\mapsto gm$, has a $G$-linear section
$$
s(m)=\sum_{g\in G/D}g\otimes\alpha(g^{-1}m).
$$
Changing a representative by an element of $D$ leaves this tensor unchanged, $G$ permutes the terms, and $\varepsilon s=\operatorname{id}_M$. Hence
$$
\boxed{M\mid\operatorname{Ind}_D^G\operatorname{Res}_D^G M.}
$$
This is <relative projectivity> for $D$, proving the <Green relative projectivity theorem>; indecomposability was needed only to speak of a single <block of a group algebra> or <vertex of an indecomposable module>.
Finally let $\hat e$ lift $e$. A trace expression for $\hat e$ reduces to one for $e$, so a modular <defect group of a block> is contained, up to conjugacy, in an integral one. Conversely choose a modular defect group $D$ and $a\in(kG)^D$ with $e=\operatorname{Tr}_D^G(a)$. Conjugation orbit sums lift $a$ to $\hat a\in(\mathcal OG)^D$. The element
$$
u=\hat e\operatorname{Tr}_D^G(\hat a)\in\hat eZ(\mathcal OG)
$$
reduces to its block identity, hence is a unit in that block center. Its inverse there is central, so
$$
\hat e=\operatorname{Tr}_D^G(u^{-1}\hat e\hat a).
$$
The integral defect is therefore contained in the modular one. \b[The corresponding blocks have the same conjugacy class of defect groups and the same defect.]
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