On parallelisms of PG(3,4) with automorphisms of order 2

Let $\mathrm{PG}(n,q)$ be the n-dimensional projective space over the finite field $F_q$. A spread in $\mathrm{PG}(n,q)$ is a set of mutually skew lines which partition the point set. A parallelism is a partition of the set of lines by spreads. The classification of parallelisms in small finite projective spaces is of interest for problems from projective geometry, design theory, network coding, error-correcting codes, and cryptography. All parallelisms of $\mathrm{PG}(3,2)$ and $\mathrm{PG}(3,3)$ are known. Parallelisms of $\mathrm{PG}(3,4)$ which are invariant under automorphisms of odd prime orders have also been classified. The present paper contributes to the classification of parallelisms in $\mathrm{PG}(3,4)$ with automorphisms of even order. We focus on cyclic groups of order four and the group of order two generated by a Baer involution. We examine invariants of the parallelisms such as the full automorphism group, the type of spreads and questions of duality. The results given in this paper show that the number of parallelisms of $\mathrm{PG}(3,4)$ is at least 8675365. Some future directions of research are outlined.