$G$ is characteristically simple $\iff$ there is simple $T$ such that $G \cong T\times T \times \cdots \times T$
Hints:
We may suppose $\;G\;$ has a non-trivial normal subgroup , otherwise the claim follows at once.
Since $\;G\;$ is finite, choose a minimal non-trivial $\;N\lhd G\;$, and look at
$$\;M:=\langle\;N^\phi\;:\;\;\phi\in\text{Aut}\,(G)\;\rangle$$
Prove now that $\;G\;$ is the direct product of some of the $\;N^\phi$'s .
Disclaimer: The only proof of the above I know is applying Zorn's Lemma on the set of $\;N^\phi$'s generating their own direct product. It seems to me weird to use this powerful weapon with a finite group, yet I cannot see right now a way out of it.