Finite skew braces with characteristically simple multiplicative group

We study finite skew braces whose multiplicative group is characteristically simple, namely of the form $S^n$ for a finite simple group $S$. Motivated by the strong rigidity phenomena known for skew braces with simple or quasisimple multiplicative group, we investigate to what extent the assumption $(B,\cdot)\cong S^n$ constrains the additive group. We first consider the two-sided case and prove that if $B$ is a finite two-sided skew brace with non-abelian characteristically simple multiplicative group, then $(B,+) \cong (B,\cdot)$. We then show that a similar rigidity phenomenon persists beyond the two-sided setting: if the additive group is also characteristically simple with the same number of direct factors, say $(B,+)\cong T^n$ for some finite simple group $T$, then necessarily $(B,+)\cong (B,\cdot)$. Next, we consider the case in which $(B,+)$ is supersolvable and $S$ is non-abelian, and prove strong restrictions: $S \cong \PSL_2(7)$, the additive group has a quotient isomorphic to $S_3$, and it contains a non-trivial elementary abelian $3$-subgroup with large centralizer. In particular, if $(B,\cdot)\cong S\times S$, then $(B,+)$ cannot be supersolvable. Finally, we investigate skew braces with characteristically simple additive group and obtain several existence and non-existence results depending on the group-theoretic type of the multiplicative group. These results exhibit a marked asymmetry: while a non-abelian characteristically simple multiplicative group imposes strong rigidity on the skew brace, the corresponding condition on the additive group leads to a substantially more flexible existence theory.