On the $H$-space of a random graph

The edge space $\mathcal{E}(G)$ of a graph $G$ is the vector space $\mathbb{F}_2^{E(G)}$ with members naturally identified with subgraphs of $G$, and the $H$-space is the subspace $\mathcal{C}_H(G)$ of $\mathcal{E}(G)$ spanned by copies of the graph $H$. We are interested in when the random graph $G = G_{n,p}$ is likely to satisfy $$\mathcal{C}_H(G) = \mathcal{W}_H(G),$$ where $\mathcal{W}_H(G)$ takes one of four natural values, depending on the value of $\mathcal{C}_H(K_n)$. We show that for strictly $2$-balanced $H$, w.h.p. the above equality holds whenever every edge of $G$ is in a copy of $H$.