TITLE: What if an automorphism fixes every maximal subgroup pointwise. Is it then the identity?
QUESTION [5 upvotes]: This question came up in the discussion over here
My first thought was that then it fixes the Frattini subgroup.   Any help?
For reference we found that the answer is no when each maximal subgroup is merely mapped back to itself.

REPLY [5 votes]: Small counterexample: $G=C_4$, the cyclic group of order 4, say generated by $a$. The automorphism $a \rightarrow  a^{-1}$, fixes  the maximal subgroup $ \langle a^2 \rangle$ pointwise.