Noetherianness is extension-closed

Statement
Suppose $$G$$ is a group and $$H$$ is a normal subgroup of $$G$$. Then, if both $$H$$ and $$G/H$$ are Noetherian groups (i.e., every subgroup of either is finitely generated), then $$G$$ is also Noetherian.

Facts used

 * 1) uses::Finite generation is extension-closed
 * 2) uses::Second isomorphism theorem

Proof
Given: $$G$$ is a group, $$H$$ is a normal subgroup of $$G$$. Both $$H$$ and $$G/H$$ are Noetherian. $$K$$ is a subgroup of $$G$$.

To prove: $$K$$ is finitely generated.

Proof: