Hurewicz theorem modulo a Serre class ($\ker h_n,\mathrm{coker}{h}_n\in \mathcal{C}$)

For a simply connected space, if lower homotopy groups lie in an admissible Serre class, the Hurewicz map in the next degree is an isomorphism modulo that class, and the lower homology groups lie in the class. This applies to finitely generated groups and to finite groups supported at prescribed primes.