Walks, Paths and Connectivity · Hard problem

Let $G$ be the "house" graph: a square on vertices $1,2,3,4$ (edges $12,23,34,41$) with an apex vertex $5$ joined to both $3$ and $4$. A trail of length $4$ from $1$ to $5$ uses $4$ distinct edges (no edge repeated), possibly revisiting a vertex. How many such trails are there?