  Let $B$ be the split extension of a finite dimensional algebra $C$ by a
$C$-$C$-bimodule $E$. We define a morphism of associative graded algebras
$\varphi^*:\HH^*(B)\rightarrow \HH^*(C)$ from the Hochschild cohomology of $B$
to that of $C$, extending similar constructions for the first cohomology groups
made and studied by Assem, Bustamante, Igusa, Redondo and Schiffler. In the
case of a trivial extension $B=C\ltimes E$, we give necessary and sufficient
conditions for each $\varphi^n$ to be surjective. We prove the surjectivity of
$\varphi^1$ for a class of trivial extensions that includes relation extensions
and hence cluster-tilted algebras. Finally, we study the kernel of $\varphi^1$
for any trivial extension, and give a more precise description of this kernel
in the case of relation extensions.

Representation Theory