Does every invertible linear map $M$ between $V \oplus V$ and $W \oplus W$ naturally yield an invertible linear map $L$ between $V$ and $W$?

Here "naturally" means "in an $GL(V) \times GL(W)$-equivariant fashion" [originally I wrote $SL(V)$ here and below, but Todd Trimble and Eric Wofsey pointed out my mistakes]. Cf. the article "Producing New Bijections From Old" by David Feldman and myself, available at http://jamespropp.org/cancel.pdf, in which we raise such questions in the context of maps between finite sets $S$ and $T$ and ask for $Sym(S) \times Sym(T)$-equivariance.

One might try to construct $L$ from $M$ by mapping $V$ to $V \oplus V$ (sending $v$ to $(v,0)$, say), mapping $V \oplus V$ to $W \oplus W$ (by $M$), and then mapping $W \oplus W$ to $W$ (sending $(w,w')$ to $w$, say). But the map $L$ constructed in this fashion need not be invertible; e.g., if $V = W$ and $M$ sends $(v,v')$ to $(v',v)$, then $L$ is the 0-map.

Something cleverer is called for; or maybe by paying attention to $GL(V)$ one can show that no such general construction is possible.

If there is an existing literature on problems of this kind, pointers would be appreciated.

(Note that Conway, Doyle, and Qiu, in http://arxiv.org/abs/math/0605779 and http://arxiv.org/abs/1504.01402, take a slightly different approach to questions of this kind. I'm not sure what a "linear" version of their approach would be.)