Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ a proper ideal of $R$. We show that if $n\,:=\,\text{grad}{{\text{e}}_{R}}\,\mathfrak{a}$, then $\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))\,\cong \,\text{Ext}_{R}^{n}(H_{\mathfrak{a}}^{n}(R),\,R)$. We also prove that, for a nonnegative integer $n$ such that $H_{\mathfrak{a}}^{i}(R)\,=\,0$ for every $i\,\ne \,n$, if $\text{Ext}_{R}^{i}({{R}_{z}},\,R)\,=\,0$ for all $i\,>\,0$ and $z\,\in \,\mathfrak{a}$, then $\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))$ is a homomorphic image of $R$, where ${{R}_{z}}$ is the ring of fractions of $R$ with respect to a multiplicatively closed subset $\{{{z}^{j}}|j\,\ge \text{0}\}$ of $R$. Moreover, if $\text{Ho}{{\text{m}}_{R}}({{R}_{z}},R)=0$ for all $z\,\in \,\mathfrak{a}$, then ${{\mu }_{H_{\mathfrak{a}}^{n}(R)}}$ is an isomorphism, where ${{\mu }_{H_{\mathfrak{a}}^{n}(R)}}$ is the canonical ring homomorphism $R\,\to \,\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))$.