Isogeny classes of non-simple abelian surfaces over finite fields

Let $A=E\times E_{ss}$ be a principally polarized almost ordinary split abelian surface over a finite field ${\mathbb{F}}_q$. We give asymptotic upper and lower bounds on the number of principally polarized abelian surfaces over ${\mathbb{F}}_{q^n}$ that are ${\bar{\mathbb{F}}}_q$-isogenous to $A$ up to isomorphism, which is a refinement of the results in the work of Achter and Howe.