Given a genus 2 curve C defined over a finite field q of odd characteristic such that 2|#Jac(C)(q), we study the growth of the 2-adic valuation of the cardinality of the Jacobian over a tower of quadratic extensions of q. In the cases of simpler regularity, we determine the exponents of the 2-Sylow subgroup of Jac(C)(q2k).
