While reading a paper, I came across the following peculiar condition:

Let $1 \rightarrow H \rightarrow G \rightarrow G/H \rightarrow 1$ be a short exact sequence, and let $H$ be abelian. We require that any automorphism, $\sigma$, of $G$ that preserves $H$ pointwise and such that $\sigma(g)H=gH$ (preserves cosets pointwise) is trivial.

This doesn't give me a good clue as to what we're talking about. For example: is it true for any semi-direct product of two groups of coprime order (the first one being abelian, as required)? How about for a semi-direct product of cyclic groups of coprime order? Prime coprime orders? What is this condition, and what are some canonical examples of it?

I apologize if this is simple, but my finite group theory is pretty shoddy.