TITLE: On finite group with inner automorphisms group
QUESTION [6 upvotes]: Let $G$ be a finite $2$-group such that $\mid Inn(G)\mid=4$ and $\Phi(G)\subsetneq Z(G)$ where $\Phi(G)$ is frattini subgroup. Then prove that there exists an $\alpha\in Aut(G)$ such that $\alpha(g)\neq g$ for some $g\in Z(G)$.
Thank you

REPLY [3 votes]: Let me give some explaination for the Derek's answer: Write $F=\Phi(G)$. Consider $\bar{G}=G/F$. Then $\bar{Z}=<\bar{x}_1, \cdots, \bar{x}_k>$ for some $x_1, \cdots, x_k\in Z(G) - F$. Then $Z=Z(G)=<x_1,\cdots, x_k, F>$. Now define a homomorphism induced by $\alpha_1: x_1\rightarrow x_1h, \cdots, x_k\rightarrow x_k, f\rightarrow f$ for $f\in F$, where $h\in F$ such that $|h|=2$. Cleary $\alpha_1$ is an isomorphism for $<\alpha_1({x_1}), \cdots, \alpha_1(x_k), \alpha_1(F)>=Z$. Now let $\bar{G}=<\bar{x}_1, \cdots, \bar{x}_k> \times <\bar{y}_1, \cdots>$. Extending $\alpha_1$ to $\alpha$ by $y_1\rightarrow y_1, \cdots $, we can get the isomorphism descripted by Derek.