We consider how Alperin's conjecture on the number of simple $kG$-modules is affected by the presence of proper, non-trivial, normal subgroups. We prove that a minimal counterexample to this conjecture has no non-trivial $p$-factor-group (where $p=\mathrm {char}(k)$), and we present other conjectures, related to Alperin's, which predict some comatibility with the action of a group of automorphisms. We prove that one of these conjectures (though apparently stronger) is actually equivalent to Alperin's conjecture.