Let $K$ be a number field. Take two representations $A$, $B$ of $G_K=\mathrm{Gal}(\bar{K}/K)$ over some ring $\Lambda$ (in fact, I only consider the case $\Lambda$ is a field, usually of characteristic 0, and $A$, $B$ are of finite dimension). Let us further suppose that $A$, $B$ ''come from geometry'' (of course, there are much well-defined notion of geometric Galois representations, but, at the moment, I do not want to dive into there. But, I usually regard ''come from geometry $\sim$ motivic''). Here is the question

Let $E$ be an extension of $A$ by $B$ in the category of $G_K$-representations. Then does $E$ also come from geometry?

Since I did not give the precise meaning of ''come from geometry'', so it is welcome to start with your definition. Thank you very much in advance.

anyalgebraic variety is always "de Rham" and hence "Hodge-Tate" at primes above $p$ -- if you google those phrases, along with "p-adic hodge theory" you should find some Galois cohomology computations showing that these properties are not closed under arbitrary extensions. IIRC there are extensions of the trivial representation by itself which are not de Rham. $\endgroup$ – Daniel Litt Aug 9 '18 at 3:01