Symmetric deformation of vector bundles

Question

Let $E_1$ and $E_2$ be two non-isomorphic extensions of vector bundles $M$ by $N$ on an algebraic variety $X$. Assume $E_1\cong E_2$. Is it possible to deform $E_1$ to $E_2$ in a symmetric way? More precisely is there a short exact sequence of vector bundle $0\rightarrow U \rightarrow V \rightarrow W \rightarrow 0$ on $X\times \mathbb{A}^1$ such that restricts to the extensions of $E_1$ and $E_2$ on $t=0,1$ and $V$ is invariant under the transformation $t\mapsto 1-t$.