If $C$ is a symmetric monoidal category, then $BC$ is canonically an algebra over a certain $E_\infty$ operad, but if $F: C \to D$ is a symmetric monoidal functor then (as far as I can see) $BF: BC \to BD$ is not a map of algebras over that operad (unless all the morphisms $(Fx) \otimes (Fy) \to F(x \otimes y)$ are identities).

Because of this I struggle to associate a spectrum map between the two spectra arising from $BC$ and $BD$ by infinite loop space theory, I can only see how to get a zig-zag where the wrong-way map is a weak equivalence. For most practical purposes that's just as good, but nevertheless I wonder: are any of the "well known" infinite loop space machines functorial on the nose, with respect to symmetric monoidal functors?