Extension-closed group property

Definition
Suppose $$\alpha$$ is a group property. We say that $$\alpha$$ is extension-closed if the following holds:

For any group $$G$$ and normal subgroup $$H$$ of $$G$$ such that both $$H$$ and the quotient group $$G/H$$ satisfy the property $$\alpha$$, $$G$$ also satisfies the property $$\alpha$$.