The nature of kappa-symmetry transformations is examined for p-branes embedded in a class of coset superspaces G/H, where G is an appropriate supergroup and H is the Lorentz subgroup. It is shown that one of the conditions delta Z(M)E(M)(a) = 0 which characterizes kappa-symmetry transformations arises very naturally if they are implemented in terms of a right action of a subgroup of the supergroup G on the supergroup elements which represent the coset. Unlike the global left action of G on G/H (which gives rise to supersymmetry on the coset superspace), then is no canonically defined right action of G on G/H. However, an interpretation of this right action involving an enlargement of the isotropy group from the Lorentz subgroup to a subgroup of G with generators which include some of the fermionic generators of G is suggested. Closure of the generators of this larger subgroup under commutation leads to the usual "brane scan" for p-branes which have bosonic degrees of freedom which are world-volume scalars. (C) 2000 Elsevier Science B.V. All rights reserved.