Solving system of equation over distinct finite fields

Question

Is it possible to solve an equation systems which involves elements from distinct finite fields? One example could be elements from $\mathrm{GF}(2)$ and $\mathrm{GF}(2^2)$:

With:
$x=[1,1,0,1]$
$y_1=A_1x$ in $\mathrm{GF}(2)$
$y_2=A_2x$ in $\mathrm{GF}(2^2)$

How to solve:
$[y_1,y_2] = [A_1,A_2]x$