Let $(X,0)$ be an ICIS of dimension $2$ and let $f :(X,0)\to\mathbb{C} ^2$ be a map germ with an isolated instability. Given $F : (\mathcal{X} , 0) \to (\mathbb{C} \times \mathbb{C}^2, 0)$ a stable unfolding of $f$, we look to the invariants related to the family $f_s$ and we find relations between them. We obtain necessary and sufficient conditions for $F$ to be Whitney equisingular. (Joint work with B. Orfice-Okamoto and J. N. Tomazella)