Forbidden triples for $2$-connected graphs with minimum degree three which contain $K_4$ and $K_{2,2}$

For a family $\mathcal{H}$ of graphs, a graph $G$ is said to be $\mathcal{H}$-free if $G$ contains no member of $\mathcal{H}$ as an induced subgraph. Let \mathcal{G}_2^{(3)}}(\mathcal{H}) denote the family of $2$-connected $\mathcal{H}$-free graphs having minimum degree at least $3$. This paper is concerned with families $\mathcal{H}$ of connected graphs with $|\mathcal{H}| = 3$ such that \mathcal{G}_2^{(3)}}(\mathcal{H}) is a finite family. In particular, we show that for a connected graph