  In this paper, for a smooth variety equiped with an action of a connected
algebraic group (not necessary linear), we introduce the notion of invariant
Brauer sub-group and the notion of invariant \'etale Brauer-Manin obstruction.
Then we prove that this obstruction is equivalent to the \'etale Brauer-Manin
obstruction. This extends the main notion and the key result of [C17]. As a
corollary, this also extends a result of B. Creutz.
