= Solution
The <decomposition matrix (modular representation theory)> separates into two connected components. The first contains $\phi_1,\phi_2$ and $\chi_1,\chi_{3A},\chi_{3B},\chi_4$; it is the principal block $B_0$. The second contains only $\phi_3$ and $\chi_5$; since $\phi_3(1)=5$ contains the full 5-part of $|A_5|=60$, this is a <defect-zero representation> and its block $B_1$ has defect group $1$.
For $D=1$, the <normalizer> is $N_G(1)=G$, so the <Brauer correspondence> is the identity and $B_1$ corresponds to itself.
The defect group of the principal block is a Sylow 5-subgroup $P\cong C_5$. There are six Sylow 5-subgroups in $A_5$, so the <orbit-stabilizer theorem> gives
$$
|N_{A_5}(P)|=\frac{60}{6}=10.
$$
The <centralizer> of a 5-cycle in $A_5$ is $P$, and an involution in the normalizer acts on $P$ by inversion. Hence
$$
N=N_{A_5}(P)\cong C_5\rtimes C_2\cong D_{10}.
$$
In characteristic five the simple $kN$-modules are inflated from $N/P\cong C_2$: their <Brauer characters> are $\psi_+=(1,1)$ and $\psi_-=(1,-1)$ on the identity and involution classes. If $1,\varepsilon,\rho_1,\rho_2$ are the two one-dimensional and two two-dimensional ordinary characters of $D_{10}$, their reductions are
$$
1\mapsto\psi_+,
\qquad
\varepsilon\mapsto\psi_-,
\qquad
\rho_1,\rho_2\mapsto\psi_++\psi_-.
$$
The resulting <decomposition matrix (modular representation theory)> is connected, so these characters form the unique 5-block $b_0$ of $N$, with defect group $P$. By the <Brauer first main theorem>,
$$
\boxed{B_0\longleftrightarrow b_0\text{ for }D=C_5,
\qquad B_1\longleftrightarrow B_1\text{ for }D=1.}
$$
This is the complete <5-modular blocks of A5> correspondence.
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