When a Pokémon's stats in Pokémon-Amie are maxed out, it gains several in-battle abilities such as increased critical-hit ratio, increased evasion, and auto-Refresh.
Will these abilities also work in online battles?
Nope, they won't work http://pokemondb.net/pokebase/163574/do-the-boosts-provided-by-pokemon-amie-go-into-wi-fi-battling <--- basically the same question