Inverse image condition

Symbol-free definition
A subgroup property is said to satisfy the inverse image condition if, for any homomorphism of groups, the inverse image of a subgroup satisfying the property in the group on the right, satisfies the property in the group on the left.

Definition with symbols
Let $$p$$ be a subgroup property. We say that $$p$$ satisfies the inverse image condition if for any homomorphism $$\phi: G \to H$$, the following holds: whenever $$N$$ is a subgroup satisfying $$p$$ in $$H$$, $$\phi^{-1}(N)$$ satisfies $$p$$ in $$G$$.

Weaker metaproperties

 * Intermediate subgroup condition
 * Transfer condition
 * Quotient-preserved subgroup property