Theorem 5.1.1 in Derek J. S. Robinson's book "A Course in the Theory of Groups" says
The class of soluble groups is closed with respect to the formation of subgroups, images, and extensions of its members.
I think this means that subgroups $H$ and quotients $G/N$ are solvable if $G$ is solvable. I don't know what the author means by "extensions of its members". Can someone please explain?
A group $G$ is called an extension of a group $Q$ by a group $N$, if $N$ is a normal subgroup of $G$ with quotient $G/N\cong Q$. In other words, we have a short exact sequence $$ 1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1. $$ Now $G$ is solvable if and only if $N$ and $G/N$ are solvable. So extensions of solvable groups are solvable. In other words, the class of solvable groups is closed under extensions of its members.
Note that the class of nilpotent groups is not closed under extensions.