Omissible extensions of SL_۲(k) where k is a field of positive characteristic

A normal subgroup N of a group G is said to be an‎ omissible subgroup of G if it has the following property‎: ‎whenever X\leq G is such that G=XN‎, ‎then G=X‎. ‎In this note we construct various groups G‎, ‎each of which has an omissible subgroup N\neq ۱ such that G/N\cong SL_۲(k) where k is a field of positive characteristic‎.

Keywords:

Omissible subgroup , special linear group , Frattini extension , locally (soluble-by-finite) group