In this paper, we develop ideas in algebraic graph theory to look at the Cayley graphs of groups, and the automorphism groups of graphs. A Cayley graph can be defined using a group and a set; for a given group, a group automorphism between the sets implies isomorphic Cayley graphs, but a graph isomorphism between two Cayley graphs does not always imply a group automorphism between the two sets. When a graph isomorphism implies a group automorphism between the sets, a Cayley graph is called a Cayley Isomorphic graph (CI-graph) and the question of when a Cayley graph is a CI-graph is an open one. In this paper, we look at when a Cayley graph with speci c automorphism groups will be a CI-graph, in particular proving that for a Cayley graph for a group G with 4k elements and automorphism group Z/4Z|Sk, the Cayley graph is a CI-graph if it has no elements of order 4, and is not a CI-graph if it contains a non-cylic subgroup of order 2^k and an element of order 4.