LTI system input upsampling
I am wondering whether it is possible to express $X_{2F_s}$ in terms of $X_{F_s}$.


In general, no.

This is the sampling theorem: unless you've bandlimited $X_{2F_s}$ first to be alias-free sampled by $F_s$, then the digital signal obtained by sampling twice as fast will contain information that the digital signal sampled at $F_s$ does not contain. And information that is not contained cannot be recovered.

If you, however, did not band limit $X$ to the Nyquist bandwidth dictated by the sampling rate, your $X_{F_s}$ will already contain aliases. Then you can simply look at the "upper" half of $X_{2F_s}$, shift it down to overlay exactly the lower half, which is exactly what's called aliasing, and reconstruct $Y_{F_s}$ from $Y_{2F_s}$.