bounds the number of communities that a member of a network can belong to. More precisely, the goal is to find in a graph $G$ at least $k$ subgraphs isomorphic to a member of $\mathcal{F}$ such that every vertex in $V(G)$ belongs to at most $\ell$ subgraphs. This type of overlap was also studied by Fellows *et al*. [@FellowsGKNU11] in the context of graph editing. Fernau *et al*. [@FernauLR15] proved, in particular, that the <span style="font-variant:small-caps;">$\mathcal{F}$-Subgraph-Packing with $\ell$-Membership</span> problem is -hard for all values of $\ell {\geqslant}1$ when ${\mathcal F}= \{F\}$ and $F$ is an arbitrary connected graph with at least three vertices, but polynomial-time solvable for smaller graphs. Note that <span style="font-variant:small-caps;">$\mathcal{F}$-Subgraph-Packing with $\ell$-Membership</span> generalizes the <span style="font-variant:small-caps;">$\mathcal{F}$-Subgraph-Packing</span> problem, which consists in finding in a graph $G$ at least $k$ vertex-disjoint subgraphs isomorphic to a member of $\mathcal{F}$. The smallest kernel for the <span style="font-variant:small-caps;">$\mathcal{F}$-Subgraph-Packing</span> problem [@Moser09] has size $O(k^{r-1})$, where $\mathcal{F} = \{F\}$ and $F$ is an arbitrary graph on $r$ vertices. A list of references of kernels for particular cases of the family $\mathcal{F}$ can be found in [@FernauLR15]. Concerning the kernelization of <span style="font-variant:small-caps;">$\mathcal{F}$-Subgraph-Packing with $\ell$-Membership</span>, Fernau *et al*. [@FernauLR15] provided a kernel on general graphs 