  We consider a curve of genus 2 defined over a $5$-adic field such that the
image of inertia of the associated Galois representation is finite and is the
largest possible, isomorphic to $C_5\rtimes C_8$. We give a few criteria to
identify such curves and prove a formula of the local root number in terms of
other invariants.
  The obtained result is analogous to a formula of root numbers of elliptic
curves, due to Kobayashi. We also present a geometric interpretation of
Kobayashi's result which eliminates explicit dependency on a particular
Weierstrass equation of a given elliptic curve.
