Let π be a field, char(π)β 2, and G a subgroup of GL(n,π). Suppose gβ¦gβ― is a π-linear antiautomorphism of G, and then define G1={gβGβ£gβ―g=I}. For C being the centraliser πG (G1) , or any subgroup of the centre π΅(G) , define G(C) ={gβGβ£gβ―gβC}. We show that G(C) is a subgroup of G, and study its structure. When C=πG (G1) , we have that G(C) =π©G (G1) , the normaliser of G1 in G. Suppose π is algebraically closed, πG (G1) consists of scalar matrices and G1 is a connected subgroup of an affine group G. Under the latter assumptions, π©G (G1) is a self-normalising subgroup of G. This holds for a number of interesting pairs (G,G1); in particular, for those that we call parabolic pairs. As well, for a certain specific setting we generalise a standard result about centres of Borel subgroups.