No. 
Shorthand 
A subgroup of a group is termed a subgroup having a transversal comprising involutions if ... 
A subgroup of a group is termed a subgroup having a transversal comprising involutions if ...

1 
left transversal comprising involutions 
it has a left transversal all of whose nonidentity elements are involutions 
there is a subset of such that intersects each left coset of at exactly one point and such that all the nonidentity elements of are involutions

2 
right transversal comprising involutions 
it has a left transversal all of whose nonidentity elements are involutions 
there is a subset of such that intersects each right coset of at exactly one point and such that all the nonidentity elements of are involutions

3 
left transversal, also right transversal, comprising involutions 
it has a left transversal that is also a right transversal, such that all the nonidentity elements are involutions 
there is a subset of such that intersects each left coset of at exactly one point, intersects each right coset of at exactly one point and such that all the nonidentity elements of are involutions.
