Let $\pi :X'\rightarrow X$ be the normalization. Let $s$ be a double point of $X$, and let $p,q$ be the two points of $\pi ^{-1}(s)$ (ordering chosen). There is an exact sequence of sheaves $$1\rightarrow \mathcal{O}_{X}^*\rightarrow \pi _*\mathcal{O}_{X'}^*\xrightarrow{\ \varphi \ } \kappa (s)^*\rightarrow 1$$here $\kappa (s)^*$ is the skyscrapper sheaf over $s$ with stalk $k^*$, and $\varphi $ maps a function $f$ to $f(p)/f(q)$. The coboundary of the associated long exact sequence gives an injective homomorphism of algebraic groups $h_s:\mathbb{G}_m\rightarrow \operatorname{Pic}^{\mathrm{o}}(X) $. If $\operatorname{Sing}(X)=\Sigma $, the homomorphisms $h_s \ (s \in \Sigma )$ define an injective homomorphism $\mathbb{G}_m^{\Sigma }\rightarrow \operatorname{Pic}^{\mathrm{o}}(X) $; if $\phi$ induces the identity on $\operatorname{Pic}^{\mathrm{o}}(X) $, it must preserve $\Sigma $ pointwise, and also $S=\pi ^{-1}(\Sigma )$ because exchanging $p$ and $q$ exchanges $h$ and $h^{-1}$.

Let $s$ be a double point of $X$, and let $p,q$ be the two points of $\pi ^{-1}(s)$ (ordering chosen). Let $\pi :X_s\rightarrow X$ be the partial normalization of $X$ at $s$. There is an exact sequence of sheaves $$1\rightarrow \mathcal{O}_{X}^*\rightarrow \pi _*\mathcal{O}_{X_s}^*\xrightarrow{\ \varphi \ } \kappa (s)^*\rightarrow 1$$here $\kappa (s)^*$ is the skyscrapper sheaf over $s$ with stalk $k^*$, and $\varphi $ maps a function $f$ to $f(p)/f(q)$. The coboundary of the associated long exact sequence gives an injective homomorphism of algebraic groups $h_s:\mathbb{G}_m\rightarrow \operatorname{Pic}^{\mathrm{o}}(X) $. If $\operatorname{Sing}(X)=\Sigma $, the homomorphisms $h_s \ (s \in \Sigma )$ define an injective homomorphism $\mathbb{G}_m^{\Sigma }\rightarrow \operatorname{Pic}^{\mathrm{o}}(X) $; if $\phi$ induces the identity on $\operatorname{Pic}^{\mathrm{o}}(X) $, it must therefore preserve $\Sigma $ pointwise, and also $S=\pi ^{-1}(\Sigma )$ because exchanging $p$ and $q$ exchanges $h$ and $h^{-1}$.