Outer linear group

In terms of the transpose-inverse map
The outer linear group of degree $$n$$ over a field $$k$$ is defined as the external semidirect product of the defining ingredient::general linear group $$GL(n,k)$$ with a cyclic group of order two, where the non-identity element of the cyclic group acts by the defining ingredient::transpose-inverse map

The definition also makes sense if the field $$k$$ is replaced by a commutative unital ring $$R$$.