  We prove that Mann-Robalo's construction of the brane action extends to
general coherent $\infty$-operads, with possibly multiple colors and
non-contractible spaces of unary operations. This requires to establish two
results regarding spaces of extensions that were left unproven in the
aforementioned construction. First, we show that Lurie's and Mann-Robalo's
models for such spaces are equivalent. Second, we prove that the space of
extensions in the sense of Lurie is not in general equivalent to the homotopy
fiber of the associated forgetful morphism, but rather to its homotopy quotient
by the $\infty$-groupoid of unary operations, correcting an oversight in
existing literature. As an application, we obtain that the $\infty$-operads of
$B$-framed little disks are coherent and therefore yield new operations on
spaces of branes of perfect derived stacks.
