Meta operator

Property-theoretic definition
The meta operator is a map from the group property space to itself, that takes as input a group property $$p$$ and outputs the square of $$p$$ under the group extension operator.

Definition with symbols
The meta operator is a map from the group property space to itself defined as follows: it takes as input a group property $$p$$ and outputs the group property $$q$$ defined as follows:

A group $$G$$ has property $$q$$ if there is a normal subgroup $$N \triangleleft G$$ such that $$N$$ and $$G/N$$ both have property $$p$$ (as abstract groups).

Application
Important instances of application of the meta operator: