$\DeclareMathOperator\Ext{Ext}$Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $F,G$ be finitely generated free $R$-modules and $f:F\to G$ be an $R$-linear map such that $f(F)\subseteq \mathfrak m G$. Let $X$ be a finitely generated $R$-module, and let $\alpha : 0\to F \to A_{\alpha} \to X \to 0$ be a short exact sequence i.e. $[\alpha]\in \Ext^1_R(X,F)$. We have a following push-out diagram with $[\beta] \in \Ext^1_R(X,G)$.

$$\require{AMScd}\begin{CD} \sigma : 0 @>>> F @>>> A_\alpha @>>> X @>>> 0 \\ @. @VVfV @VVV @| \\ \beta : 0 @>>> G @>>> A_\beta @>>> X @>>> 0. \end{CD}$$ My question is: must it be necessarily true that $[\beta] \in \mathfrak m \Ext^1_R(X,G)$?

Some thoughts: The answer is affirmative if $F\cong G\cong R$. Indeed, in this case, $f:R\to R$ must be given by multiplication by some $x\in R$. Since $f(F)\subseteq \mathfrak m G$, so $x\in \mathfrak m$. Then in $[\beta]=x[\alpha]\in \Ext^1_R(X,R)$, hence $[\beta]\in x\Ext^1_R(X,R)\subseteq \mathfrak m \Ext^1_R(X,G) $.