Antitransitive subgroup property

Symbol-free definition
A subgroup property is said to be antitransitive if its conjunction with the square of its proper part is the fallacy.

Definition with symbols
A subgroup property $$p$$ is said to be antitransitive if whenever $$H < K < G$$ are groups with all the inclusions strict, such that $$H$$ satisfies $$p$$ in $$K$$ and $$K$$ satisfies $$p$$ in $$G$$, then $$H$$ cannot satisfy $$p$$ in $$G$$.

Stronger metaproperties

 * Left-antihereditary subgroup property
 * Right-antihereditary subgroup property

Relation with transitivity
A subgroup property can be both transitive and antitransitive if and only if the square of its proper part is the fallacy, or equivalently, its proper part is a nil squareroot.