Let $G$ is finite and
$$\frac{G}{Z(G)}\cong \langle aZ(G), bZ(G)\mid a^{3}, b^{2}, (ab)^2\in Z(G)\rangle $$
and $Z(G)$ is not trivial. Then does there exist an automorphism $\alpha$ of $G$ such that $C(\alpha)\neq C(\beta)$ for any $\beta\in Inn(G)$ wherein $C(\alpha)=\{g\in G\mid \alpha(g)=g\}$?
For example i know if $2$ is a divisor of order $Z(G)$ , then this is true. Since $\mid Z(G)\mid$ is even, then there exixt $x\in Z(G)$ such that order $x$ is 2. Now if we define $\alpha(a)=a$ and $\alpha(b)=bx$ and $\alpha(g)=g$ for all $g\in Z(G) $, then $C(\alpha)\neq C(\beta)$ for any $\beta\in Inn(G)$. Thanks
