  It is shown that the gauge invariance and gauge dependence properties of
effective action for Yang-Mills theories should be considered as two
independent issues in the background field formalism. Application of this
formalism to formulate the functional renormalization group approach is
discussed. It is proven that there is a possibility to construct the
corresponding average effective action invariant under the gauge
transformations of background vector field. Nevertheless, being gauge invariant
this action remains gauge dependent on-shell.
