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Let $\mathcal{R}$ be a free Lie conformal algebra of rank $2$ with$\mathbb{C}[\partial]$-basis $\{L,I\}$ and relations \begin{eqnarray*}\left[L_{\lambda} L\right]=(\partial+2 \lambda) (L+I),\ \left[L_{\lambda}I\right]=(\partial+\lambda) I, \ \left[I_{\lambda} L\right]=\lambda I,\\left[I_{\lambda} I\right]=0. \end{eqnarray*} In this paper, we first classifyall finite nontrivial irreducible conformal modules over $\mathcal{R}$. Then wedetermine extensions between two finite irreducible conformal$\mathcal{R}$-modules.

Conformal modules and their extensions of a Lie conformal algebra related to a 2-dimensional Novikov algebra