  Let $p$ be a prime, $G$ a $p$-group, and $k$ a field of characteristic $p$.
We give a complete description of the group $\mathcal{E}_k(G)$ of endotrivial
complexes, identifying $\mathcal{E}_k(G)$ with the additive subgroup of
integer-valued class functions satisfying the Borel-Smith conditions. This is
done by constructing a short exact sequence isomorphic to a short exact
sequence of rational $p$-biset functors constructed by Bouc and Yal\c{c}in,
endowing $\mathcal{E}_k$ with rational $p$-biset structure. We also determine
all the possible h-marks which arise in $\mathcal{E}_k(G)$ when $G$ is a finite
group via the induced restriction map to a Sylow $p$-subgroup. As a
consequence, we prove that every $p$-permutation autoequivalence of a $p$-group
extends to a splendid Rickard autoequivalence. Additionally, we give a positive
answer to a question of Gelvin and Yal\c{c}in, showing the kernel of the Bouc
homomorphism for an arbitrary finite group $G$ is described by integer-valued
class functions on $p$-subgroups satisfying the oriented Artin-Borel-Smith
conditions.
