What if an automorphism fixes every maximal subgroup pointwise. Is it then the identity?

Question

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.

Answer 1

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.

Answer 2

The additive group of the rational numbers has no maximal subgroups, yet it admits many non-trivial automorphisms $x \mapsto kx$ for fixed $k \ne 0$.