Comparison of the Discrete and Continuous Cohomology Groups of a Pro-$p$ Group

We address the following question. For which finitely generated pro-$p$ groups the comparison map $\phi^2:H_{cont}^{2}(P,\F_p) \to H_{disc}{2}(P,\F_p)$ is an isomorphism? We prove that if $P$ is not finitely presented then $\phi^2$ is not surjective. Furthermore, if $P$ is finitely presented $\phi^2$ is an isomorphism if and only if the comparison map $\phi_2:H^{disc}_{2}(P, \F_p) \to H^{cont}_{2}(P, \F_p)$ of second homology groups is an isomorphism. This is the content of Theorem A. The second main result of the paper is Theorem B, which gives an explicit construction of a cochain from the kernel of $\phi^2$.