Parallelisms of PG(3,4) invariant under an elementary abelian group of order 4

This paper is a contribution to the classification of parallelisms in three-dimensional projective spaces over small finite fields of order q by computer. The smallest space in which parallelisms have not yet been classified is for \(q=4.\) Partial results are available. The parallelisms admitting a nontrivial automorphism of odd prime order are known. Moreover, much is known about the case of parallelisms of \({{\mathrm{PG}}}(3,4)\) whose automorphism group is a two group. Namely, everything is known for two of the three possible groups of order two, as well as for cyclic groups of order 4. The present paper will settle the case of parallelisms whose automorphism group is elementary abelian of order 4. This leaves open the cases of parallelisms whose full automorphism groups are either trivial or a specific group of order two.