Extending involutions on Frobenius algebras

Let A be a central simple algebra of degree n over a field of characteristic different from 2 and let B subset of A be a maximal commutative subalgebra. We show that if there is an involution on A that preserves B and such that the socle of each local component of B is a homogeneous C-2-module for this action, then B is a Frobenius algebra. For a fixed commutative Frobenius algebra B of finite dimension n equipped with an involution sigma, we characterize the central simple algebras A of degree n that contain B and carry involutions extending sigma.