Riemannian geometry, symmetry groups are subgroups of $\text{Diff}(X)$. Instead, in the *synthetic* context of metric-measure spaces, symmetry groups are subgroups of $\text{Aut}(X)$, the group of automorphisms or bijections of $X$: let $H$ be a subgroup of $\text{Aut}_{\mu}(X)$, the group of $\mu$-preserving automorphisms of $X$; denote by $M^{p}_{1,H}(X,\mu)$ the subspace of $M^{p}_1(X,\mu)$ made up of $H$-invariant functions. The main result in this work is:

\[main2\] Assume that $(X, d, \mu)$ is a metric-measure space that is compact, Ahlfors lower $s$-regular, with $(X,d)$ doubling, and such that $M^p_1(X,\mu)$ is reflexive. If $H$ is a subgroup of $\text{Aut}_{\mu}(X)$ such that for every $x$ in $X$ the set $H(x)$ is uncountable, then $$\label{comp-emb-3}
M^p_{1,H} (X,\mu) \hookrightarrow \hookrightarrow L^{q} (X,\mu)$$ whenever $1 < p < s$ and $1 \leq q \leq p^{*} = \frac{s p}{s - p}$.

In contrast with classical Sobolev spaces, there are situations where $M^p_1(X,\mu)$ is not reflexive for $1<p<\infty$: some examples of this unexpected phenomena are self similar Cantor sets, see [@rissanen]. On the other hand, a discussion about sufficient conditions on $(X,d,\mu)$ for $M^p_1(X,\mu)$ to be reflexive can be found in [@gorka1; @Hajlasz1].[^5]

To highlight the contributions of this work, we make some remarks:

1.  Concerning the groups appearing in 