= Blocks and Cartan matrices under central p-quotients
Let $Z=\langle z\rangle\leq Z(G)$ have order $p$ and set $t=z-1$. The <nilpotent ideal> $t kG$ is generated by a central element. If an idempotent lifts a central idempotent of the quotient, its off-diagonal corner satisfies $eA(1-e)\subseteq t eA(1-e)$ and therefore is zero. Thus central block idempotents lift centrally and uniquely. A <principal indecomposable module> is free over $kZ$, and the $p$ factors of its $t$-filtration are all its reduction modulo $t$. Consequently, under the natural indexing of simple modules, $C_G=pC_{G/Z}$.
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