Each p-block of a finite group has an associated defect group, which is a p-subgroup of the group. Each real 2-block has, in addition, an associated extended defect group, which is a 2-subgroup of the group that contains a defect group as a subgroup of index at most 2. We consider the possible extended defect groups of a real 2-block that has a cyclic or a Klein-four defect group. In each case we describe the modules in the block that are components in the permutation module of the group acting by conjugation on its involutions. We also determine the Frobenius-Schur indicators of the irreducible characters in the block.