Let $K = \mathbb{Q}_p$, $G = G_K$ its absolute Galois group. Let $$1\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 1$$ be a split exact sequence of (not necessarily abelian) group schemes over $K$, equipped with a $G$ action. The exact sequence is $G$-equivariant.

Let $B_{cr}$ denote the crystalline period ring, and consider the commutative diagram of Galois cohomology sets: $$H^0(G,C)\longrightarrow H^1(G,A)\longrightarrow H^1(G,B)\longrightarrow H^1(G,C)\\ \downarrow\quad\quad\quad\quad\quad\quad\downarrow\quad\quad\quad\quad\quad\quad\downarrow\quad\quad\quad\quad\quad\quad\downarrow\\ H^0(G,C(B_{cr}))\longrightarrow H^1(G,A(B_{cr}))\longrightarrow H^1(G,B(B_{cr}))\longrightarrow H^1(G,C(B_{cr})). $$ Denote by $H^1_f(G, B)$ the kernel of the vertical arrow $H^1(G,B)\longrightarrow H^1(G,B(B_{cr}))$, and we define $H^1_f(G,S), H^1_f(G,A)$ in a similar way (as the kernel of the corresponding vertical arrow).

The commutativity of the diagram, together with the exactness of the rows, implies that the restriction of the horizontal morphism $\pi: H^1(G,B)\longrightarrow H^1(G,C)$ to $H^1_f(G,B)$ lands in $H^1_f(G,C)$. Now, let $[c]\in H^1_f(G,B)$ be a cocycle class. I want to try an express the crystalline fibre of the map $\pi$ over $\pi([c])$, i.e., I want to express $\pi^{-1}(\pi([c]))\cap H^1_f(G,B)$ in terms of cohomology sets of certain Serre twists.

In Serre's "Galois Cohomology", he says that the fibre $\pi^{-1}(\pi([c]))$, is in bijection with the set $H^1(G,{}_cA)/H^0(G,{}_cC)$, where ${}_cA$ is the Serre twist of $A$ by $c$, which, as an algebraic group scheme is isomorphic to $A$, but the Galois action is twisted by $c$ as follows. If $\widetilde{a}$ is the element of ${}_cA$ corresponding to $a\in A$, then ${}^g\widetilde{a} = c(g)^{-1}\widetilde{{}^ga}c(g)$, where $\widetilde{{}^ga}$ is the element of ${}_cA$ that corresponds to the element ${}^ga\in A$, and ${}_cC$ is defined similarly.

Denote the restriction of $\pi$ to $H^1_f(G,B)$ by $\pi_f$, then I imagine that $\pi^{-1}_f(\pi_f([c]))$ is bijective with the fibred product: $$H^1(G,{}_cA)/H^0(G,{}_cC)\times_{H^1(G, B(B_{cr}))} *,$$

where $*$ is trivial cocycle of $H^1(G,B)$.

My question is if, similar to the non-crystalline characterization of Serre, there is a bijection between the set $\pi^{-1}_f(\pi_f([c]))$ and $H^1_f(G,{}_cA)/H^0_f(G,{}_cC)$.

Thanks in advance!

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