Solvability is extension-closed

Statement
Suppose $$G$$ is a group with a normal subgroup $$H$$, so that $$G/H$$ is the quotient group. Suppose, further, that both $$H$$ and $$G/H$$ are solvable groups. Then, $$G$$ is also a solvable group.

Moreover, the derived length of $$G$$ is at most equal to the sum of the derived lengths of $$H$$ and $$G/H$$.

Converse

 * Solvability is subgroup-closed
 * Solvability is quotient-closed
 * Solvability is quasivarietal (implies both being subgroup-closed and quotient-closed).

Opposite facts

 * Nilpotency is quasivarietal

Proof based on the subnormal series definition
The idea is as follows: we combine the subnormal series for $$H$$ with a series from $$H$$ to $$G$$ obtained by taking the inverse image of the subnormal series for $$G/H$$. The fourth isomorphism theorem (or more specifically, the fact that normality satisfies inverse image condition) guarantees that the inverse image of the subnormal series also satisfies the criterion that each member is adjacent in the next.