The socle is characteristic and decomposes as a direct product of minimal normal subgroups

By [A1], the subgroup generated by all minimal normal subgroups of $G$ is stable under every automorphism of $G$. By definition this subgroup is $\mathrm{soc}(G)$, so $\mathrm{soc}(G)$ is characteristic in $G$.

Let $N_1,\dots ,N_r$ be a maximal family of pairwise distinct minimal normal subgroups of $G$ chosen so that none is contained in the product of the preceding ones; when $G=1$, this family is empty. For $i\ge 1$, [L1] shows that $N_i$ centralizes $N_1\cdots N_{i-1}$, and minimality gives $N_i\cap (N_1\cdots N_{i-1})=1$ because the intersection is a normal subgroup of $G$ contained in $N_i$ but $N_i$ was chosen outside the preceding product. Hence $N_1\cdots N_r$ is an internal direct product, with the case $r=0$ giving the trivial group.

The product $N_1\cdots N_r$ is generated by minimal normal subgroups, so it lies in $\mathrm{soc}(G)$. Conversely, if $M$ is any minimal normal subgroup of $G$ not contained in $N_1\cdots N_r$, then adjoining $M$ would contradict maximality of the chosen family; therefore every minimal normal subgroup of $G$ lies in the displayed product. Thus $\mathrm{soc}(G)=N_1\times \cdots \times N_r$. By [L3], each nontrivial factor $N_i$ is characteristically simple, and [L2] then makes it a direct product of isomorphic simple groups.