Let \(k\) be a global field; let \(k'\) be a quadratic extension of \(k\). The author considers in this paper inner forms \(G\) of \(\text{GL} (2n)/k\) along with certain inner forms \(C\) of \(\text{GL} (n)/k'\) which are subgroups of \(G\). Let \(\pi\) be an automorphic representation of \(G(k_A)\); we say that \(\pi\) is cyclic if the period with respect to \(C\) does not vanish. The author defines a notion of being ‘more split’ for the algebraic groups \(G\). Given a cuspidal \(\pi\) for \(G\) and an inner form \(G'\) which is more split than \(G\), Flicker and Kazhdan have proven, under certain circumstances, that there is a ‘corresponding’ representation \(\pi'\) of \(G'(k_A)\). The purpose of this paper is to provide evidence for the statement that, if \(\pi\) is cyclic, then \(\pi'\) is also cyclic with respect to a certain \(C'\). This would cover a number of known statements.