  Let $\Gamma$ be a fractal $h$-set and ${\mathbb{B}}^{{\sigma}}_{p,q}(\Gamma)$
be a trace space of Besov type defined on $\Gamma$. While we dealt in [9] with
growth envelopes of such spaces mainly and investigated the existence of traces
in detail in [12], we now study continuous embeddings between different spaces
of that type on $\Gamma$. We obtain necessary and sufficient conditions for
such an embedding to hold, and can prove in some cases complete
characterisations. It also includes the situation when the target space is of
type $L_r(\Gamma)$ and, as a by-product, under mild assumptions on the $h$-set
$\Gamma$ we obtain the exact conditions on $\sigma$, $p$ and $q$ for which the
trace space ${\mathbb{B}}^{{\sigma}}_{p,q}(\Gamma)$ exists. We can also refine
some embedding results for spaces of generalised smoothness on $\mathbb R^n$.
