I have been studying semidirect product of groups these many days. I came to know that for every group $G$, there is a group $Hol(G) = G \rtimes_{f} Aut(G)$ (here, $f: Aut(G) \rightarrow Aut(G)$ is the identity map), called the holomorph of the group $G$, such that every automorphism of $G$ can be realised as an inner automorphism inside $Hol(G)$.
Now, if $A_{5}$ denote the alternating group on $5$ letters, then I can see that $Hol(A_{5}) \cong A_{5} \rtimes_{t} S_{5}$, for some $t$. After that, I was thinking about the socle of the Holomorph of the group $A_{5}$, it seems that its socle is contained in $A_{5} \rtimes_{t} A_{5}$, but I can't see how to prove it, or whether it is true. Can anyone tell me anything about it?