  We study a free transmission problem driven by degenerate fully nonlinear
operators. Our first result concerns the existence of solutions to the
associated Dirichlet problem. By framing the equation in the context of
viscosity inequalities, we produce optimal regularity results for viscosity
solutions and certain strong solutions to the problem. Our findings include
regularity in $ C^{1,\alpha}$ spaces, and an explicit characterization of
$\alpha$ in terms of the degeneracy rates. We argue by perturbation methods,
relating our problem to a homogeneous, fully nonlinear uniformly elliptic
equation.
