Automorphisms of models of set theory and extensions of NFU

In this paper we exploit the structural properties of standard and non-standard models of set theory to produce models of set theory admitting automorphisms that are well-behaved along an initial segment of their ordinals. $\mathrm{NFU}$ is Ronald Jensen's modifcation of Quine's `New Foundations' set theory that allows non-sets into the domain of discourse. The axioms $\mathrm{AxCount}$, $\mathrm{AxCount}_\leq$ and $\mathrm{AxCount}_\geq$ each extend $\mathrm{NFU}$ by placing restrictions on the cardinality of a finite set of singletons relative to the cardinality of its union. Using the results about automorphisms of models of set theory we separate the consistency strengths of these three extensions of $\mathrm{NFU}$. We show that $\mathrm{NFU} + \mathrm{AxCount}$ proves the consistency of $\mathrm{NFU} + \mathrm{AxCount}_\leq$, and $\mathrm{NFU} + \mathrm{AxCount}_\leq$ proves the consistency of $\mathrm{NFU} + \mathrm{AxCount}_\geq$.