Odd Order Products of Conjugate Involutions in Linear Groups over GF(2^a)

Let $G$ be isomorphic to $GL_n(q)$, $SL_n(q)$, $PGL_n(q)$ or $PSL_n(q)$, where $q=2^a$. If $t$ is an involution lying in a $G$-conjugacy class $X$, then for arbitrary $n$ we show that as $q$ becomes large, the proportion of elements of $X$ which have odd-order product with $t$ tends to $1$. Furthermore, for $n$ at most $4$ we give formulae for the number of elements in $X$ which have odd-order product with $t$, in terms of $q$