Linear symmetries of Boolean functions

We give a necessary and sufficient condition that a subgroup of GLn(2) is linearly representable and obtain some results on linear representability of its subgroups. Our results generalize some theorems from P. Clote and E. Kranakis [SIAM J. Comput. 20 (1991) 553-590]; A. Kisielewicz [J. Algebra 199 (1998) 379-403].