Let \(o\) be the ring of integers in a finite extension field of \(\mathbb{Q}_p\). Let \(k\) be its residue field. Let \(G\) be a split reductive group over \(\mathbb{Q}_p\). Let \(T\) be a maximal split torus in \(G\). Let \(I_0\) be a pro-\(p\)-Iwahori subgroup fixing a chamber \(C\) in the \(T\)-stable apartment of the semisimple Bruhat Tits building of \(G\). Let \(H(G,I_0)\) be the pro-\(p\)-Iwahori Hecke \(o\)-algebra. Let \(\operatorname{Mod}^{\operatorname{fin}}(H(G,I_0))\) denote the category of \(H(G,I_0)\)-modules of finite \(o\)-length. From a certain additional datum \((C^{(\bullet)},\phi,\tau)\) the author constructed in [Duke Math. J. 165, No. 8, 1529–1595 (2016; Zbl 1364.11103)] an exact functor \(M\mapsto D(\Theta_\ast,\mathcal{V}_M)\) from \(\operatorname{Mod}^{\operatorname{fin}}(H(G,I_0))\) to the category of étale \((\varphi^r,\Gamma)\)-modules (with \(r\in\mathbb{N}\)) depending on \(\varphi\). For \(G=\operatorname{GL}(2,\mathbb{Q}_p)\), when precomposed with the functor of taking \(I_0\)-invariants, this yields the functor from [smooth \(o\)-torsion representations of \(\operatorname{GL}(2,\mathbb{Q}_p)\) (or at least from those generated by their \(I_0\)-invariants)] to [étale \((\varphi,\Gamma)\)-modules] that plays a crucial role in Colmez’ construction of a \(p\)-adic local Langlands correspondence for \(\operatorname{GL}(2,\mathbb{Q}_p)\). In [loc. cit.] the author studied in detail the functor \(M\mapsto D(\Theta_\ast,\mathcal{V}_M)\) when \(G=\operatorname{GL}(d+1,\mathbb{Q}_p)\) for \(d\ge 1\). In [K. Kozioł, Int. Math. Res. Not. 2016, No. 4, 1090–1125 (2016; Zbl 1347.11048)] the situation has been analyzed for \(G=\operatorname{SL}(d+1,\mathbb{Q}_p)\). The purpose of this paper is to explain how the general construction of [the author, loc. cit.] can be employed concretely for other classical matrix groups \(G\), as well as for \(G\)’s of type \(E_6\), \(E_7\).