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The decomposition matrix separates into two connected components. The first contains ${\varphi }_1,{\varphi }_2$ and ${\chi }_1,{\chi }_{3A},{\chi }_{3B},{\chi }_4$; it is the principal block $B_0$. The second contains only ${\varphi }_3$ and ${\chi }_5$; since ${\varphi }_3(1)=5$ 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 givesThe centralizer of a 5-cycle in $A_5$ is $P$, and an involution in the normalizer acts on $P$ by inversion. HenceIn 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 $1,\epsilon ,{\rho }_1,{\rho }_2$ are the two one-dimensional and two two-dimensional ordinary characters of $D_{10}$, their reductions areThe resulting decomposition matrix is connected, so these characters form the unique 5-block $b_0$ of $N$, with defect group $P$. By the Brauer first main theorem,This is the complete 5-modular blocks of A5 correspondence.