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provided the initial project direction with the construction of group invariant Butson Hadamard matrices by using translation planes, Theorem 3.1.  We expanded upon his ideas which resulted in the nested cyclic subgroups construction, recursive hyperplane construction, Theorems 4.2 and 5.2. I proved the correctness of the constructions.  Prof. Xiang gave the construction of group invariant Butson Hadamard matrices using Galois Rings, Theorem 6.2.  I discovered that the folding theorem can be applied to our construction to obtain more group invariant Butson Hadamard matrices. Prof. Bernhard refined the idea which resulted in the folding construction, Theorem 7.3.  The necessary conditions on the sum of roots of unity, Theorem 2.3, were first formulated by Prof. Schmidt. Together we proved the result.