In this communication we develop a new full Slepian-Bangs formula adapted to observations on Lie Groups (LGs) distributed according to a Gaussian distribution on LGs (CGD), where both the LG mean and LG covariance depend on an unknown LG parameter. This formulation, which can be seen as a generalization on LGs of the full Slepian-Bangs formula for the Euclidean Gaussian distribution, is obtained using LG tools and properties of the CGD. A closed-form expression is then obtained for a modified Wahba's problem where the observations' covariance matrix depends on the unknown rotation matrix. Such expression is validated through numerical simulations.