# Is there an action of a group on the set of its finite quotients?

Is there an action (which turns out to be useful) of a group $G$ on the set (or isomorphism classes) of its finite quotients?

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Does it have to be the group $G$ itself? $\text{Out}(G)$ acts on the set of finite-index normal subgroups in an obvious way. – Qiaochu Yuan Aug 5 '11 at 1:45