Let $G=H\times K$ and $H\times 1$ be a characteristic subgroup of $G$.
Then can we conclude that $1\times K$ is also a characteristic subgroup of $G$?
My motivation is the case where orders of $H$ and $K$ are relatively prime. In that case, both must be characteristic subgroups of $G$. So I wonder: if one of the components is characteristic in $G$ then is the other, too?