Let $H$ be an invariant subgroup of some group $G$. Is it then true that the cosets of $H$ consist entirely of conjugacy classes?
The question was prompted by something I was thinking about in representation theory. As above, suppose $H$ is an invariant subgroup. Then we can consider the quotient group $F=G/H$. We note that $f: G \to F$ defined by $f(g)=gH$ is a homomorphism.
Now suppose $T$ is a representation of $F$. Then the composition $T \circ f$ is a representation of $G$. Under this representation, the character of some $g \in G$ is simply the character of $T(gH)$, so all elements in a coset $gH$ have the same character. We also know that elements in the same conjugacy class always have the same character. Hence we might suspect that a conjugacy class $C$ is contained entirely in a single coset. Does this have to be the case?