Let $V$ and $W$ be isomorphic vector spaces (not necessarily finite dimensional) over the same field ($\mathbb{R}$ or $\mathbb{C}$) via a linear isomorphism $f:V\to W$. Suppose $V_1, V_2$ are linear subspaces of $V$, and $W_1, W_2$ are linear subspaces of $W$ so that $$V=V_1\oplus V_2,\qquad W=W_1\oplus W_2.$$ Suppose $V_1$ and $W_1$ are isomorphic via a linear isomorphism $g:V_1\to W_1$. We know that $V_2$ and $W_2$ are not isomorphic in general, even though $V_2\cong V/V_1$ and $W_2\cong W/W_1$.
My question is, is there any criterion so that $V_2\cong W_2$? That is, any criterion which makes $V\cong W$ and $V_1\cong W_1$ imply $V/V_1\cong W/W_1$?
Edit: I should mention that, in the case I am considering, $V$, $V_1$, $W$ and $W_1$ are infinite dimensional, and $V_2$ and $W_2$ are finite dimensional.