The plethysms of the Weyl characters associated to a classical Lie group by
the symmetric functions stabilize in large rank. In the case of a power sum
plethysm, we prove that the coefficients of the decomposition of this
stabilized form on the basis of Weyl characters are branching coefficients
which can be determined by a simple algorithm. This generalizes in particular
some classical results by Littlewood on the power sum plethysms of Schur
functions. We also establish explicit formulas for the outer multiplicities
appearing in the decomposition of the tensor square of any irreducible finite
dimensional module into its symmetric and antisymmetric parts. These
multiplicities can notably be expressed in terms of the Littlewood-Richardson
coefficients