The present work develops certain analytical tools required to construct and compute invariant kernels on the space of complex covariance matrices. The main result is the $\mathrm{L}^1\,-$Godement theorem, which states that any invariant kernel, which is (in a certain natural sense) also integrable, can be computed by taking the inverse spherical transform of a positive function. General expressions for inverse spherical transforms are then provided, which can be used to explore new families of invariant kernels, at a rather moderate computational cost. A further, alternative approach for constructing new invariant kernels is also introduced, based on Ramanujan's master theorem for symmetric cones.