This talk is the second in a series of three presentations (together with Primo\v{z} Poto\v{c}nik and Gabriel Verret) concerning the Weiss conjecture and its generalisations. Let $\Gamma$ be a connected $G$-vertex-transitive graph, $v$ be a vertex of $\Gamma$ and $L$ be the permutation group induced by the vertex stabiliser $G_v$ on the neighbourhood $\Gamma(v)$ of $v$. The pair $(\Gamma,G)$ is said to be {\em locally}-$L$. A transitive permutation group $L$ is graph-restrictive if there exists a constant $c(L)$ such that, for every locally-$L$ pair $(\Gamma,G)$ and an arc $(u,v)$ of $\Gamma$, the size of the arc stabiliser $G_{u,v}$ is at most $c(L)$. In the first part of this talk, we give some details of the proof that, if $L$ is graph-restrictive, then $L$ is semiprimitive (that is, every normal subgroup of $L$ is either transitive or semiregular). This gives a positive solution to one of the implications of the main conjecture given in the first talk of this series (that is, $L$ is graph-restrictive if and only if $L$ is semiprimitive). In the second part of this talk, we show that the other implication of the conjecture (and in particular the Weiss conjecture) holds true under the mild hypothesis that the group $G$ has composition factors of bounded rank.