Let $g_\mu:\mathbb R\to \mathbb R$ be a family of smooth maps with bifurcation parameter $\mu\in \mathbb R$. I want to show that
if $g_\mu^2$ has a saddle node bifurcation as $\mu$ passes through $\mu_*$ and $g^2_\mu$ has a unique fixed point $x_*$ at $\mu=\mu_*$, then the 2 fixed points of $g_\mu^2$ that emerge from the saddle-node bifurcation at $\mu_*$ must also be fixed points of $g_\mu$.
My idea: For $\mu>\mu^*$, let $\{a,b\}\subset \mathbb R$ be the two distinct fixed points that emerge from the saddle-node bifurcation at $\mu_*$. Since $g_\mu^2(a)=a$ and $g^2_\mu(b)=b$, $\{a,b,g_\mu(a),g_\mu(b)\}$ are fixed points of $g_\mu^2$. By the fact that $P^*$ is the only fixed point of $g^2_\mu$ at $\mu=\mu^*$, we get that $g^2_\mu$ has only 2 distinct fixed points. In particular, $$\{a,b,g_\mu(a),g_\mu(b)\}=\{a,b\} \quad \iff \quad g_\mu(a)= a\, \& g_\mu(b)= b \quad or \quad g_\mu(a)=b \, \& \, g_\mu(b)=a$$ I'm not complitely sure of why the case with $g_\mu(a)=b \, \& \, g_\mu(b)=a$ is not possible. Is something related to the smoothness of $g_\mu$ at $\mu=\mu^*$?