Let $G$ be a $2$-connected $3$-regular graph. Is it true that $E(G) = E_1 \cup E_2$ where $G[E_1]$(the induced subgraph on $E_1$) is a cycle of $G$ and $G[E_2]$ is a forest (Acyclic subgraph) of $G$?

**Edit:**

Since the counterexamples presented so far are $2$-connected, what happen if the connectivity of the graph is $3$ instead of two?