Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called a class-preserving if for each $x\in G$, there exists an element $g_x\in G$ such that $\alpha(x)=g_x^{-1}xg_x$. An automorphism $\alpha$ of $G$ is called a $2$nd class-preserving if for each $x\in G$, there exists an element $g_x\in G'=[G,G]$ such that $\alpha(x)=g_x^{-1}xg_x$. Let $\mathrm{Aut_c}(G)$ and $\mathrm{Aut_c^2}(G)$ respectively denote the group of all class-preserving and $2$nd class-preserving automorphisms of $G$.

I have made a GAP program to find the structure of $\mathrm{Aut_c}(G)$ but I failed to make a GAP program to find the structure of $\mathrm{Aut_c^2}(G)$. My question is the following:

Can anybody help me to make a GAP program to find the structure of $\mathrm{Aut_c^2}(G)$?