Let q be a power of a prime p and let G denote the general linear group \(GL(n,q^ 2)\) of degree n over \(GF(q^ 2)\). Let U denote the unitary group \(U(n,q^ 2)\) of degree n over \(GF(q^ 2)\) and M the general linear group GL(n,q) over GF(q). We show that there are one-to-one correspondences between the U,U-double cosets in G and the conjugacy classes in M and between the M,M-double cosets in G and the conjugacy classes in U. We are then able to show that the Hecke algebras associated with the permutation characters \(1_ U^{ G}\) and \(1_ M^{ G}\) are commutative, which implies that these characters are multiplicity-free. Let F be the automorphism of \(GF(q^ 2)\) given by \(F(x)=x^ q\) and let F also denote the corresponding Frobenius map of G. We show that the irreducible constituents of \(1_ U^{ G}\) are precisely the irreducible characters fixed by F. Similarly, the irreducible constituents of \(1_ M^{ G}\) are precisely the irreducible characters of G fixed by the twisted Frobenius map \(F^*\). The main tool in our proof is a well-known theorem of Lang on algebraic groups.