  Let $\mathcal{W}(b)$ be a class of free Lie conformal algebras of rank $2$
with $\mathbb{C}[\partial]$-basis ${L,H}$ and relations \begin{eqnarray*}
[L_\lambda L]=(\partial+2\lambda)L,\ \ [L_\lambda
H]=\big(\partial+(1-b)\lambda\big)H, \ \ [H_\lambda H]=0, \end{eqnarray*} where
$b$ is a nonzero complex number. In this paper, we classify extensions between
two finite irreducible conformal modules over the Lie conformal algebras
$\mathcal{W}(b)$.
