Let $M$ be a non-zero finitely generated module over a commutative Noetherian local ring $(R,\mathfrak{m})$ with $\dim_R(M)=t$. Let $I$ be an ideal of $R$ with $grade(I,M)=c$. In this article we will investigate several natural homomorphisms of local cohomology modules. The main purpose of this article is to investigate when the natural homomorphisms $\gamma: Tor^{R}_c(k,H^c_I(M))\to k\otimes_R M$ and $\eta: Ext^{d}_R(k,H^c_I(M))\to Ext^{t}_R(k, M)$ are non-zero where $d:=t-c$. In fact for a Cohen-Macaulay module $M$ we will show that the homomorphism $\eta$ is injective (resp. surjective) if and only if the homomorphism $H^{d}_{\mathfrak{m}}(H^c_{I}(M))\to H^t_{\mathfrak{m}}(M)$ is injective (resp. surjective) under the additional assumption of vanishing of Ext modules. The similar results are obtained for the homomorphism $\gamma$. Moreover we will construct the natural homomorphism $Tor^{R}_c(k, H^c_I(M))\to Tor^{R}_c(k, H^c_J(M))$ for the ideals $J\subseteq I$ with $c = grade(I,M)= grade(J,M)$. There are several sufficient conditions on $I$ and $J$ to provide this homomorphism is an isomorphism.

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