My question is related to the following question by Mark Grant here on math overflow:

Formal group law of unoriented cobordism

There it is stated that $MO_*$ has a formal group law $F_0$, universal for formal group laws in characteristic $2$ which satisfy $F(x,x) = 0$.

Of course this formal group law is classified by a map $MU_* \to MO_*$ and in view of the model of $MU$ as unitary ring spectrum and $MO$ as orthogonal ring spectrum this map is indeed induced by a canonical map of (unitary) ring spectra $$c: MU \to MO,$$

see also the answer by Neil Strickland here: Which cohomology theories are real- and complex-orientable? .

Now I have two questions:

Q1) Does the map $c$ factor over $MSO$ ?

Q2) If so, can one say anything about the corresponding formal group law over $MSO$?

In view of how I think one can construct the map $MU \to MO$ I think it should be obvious that the map factors through $MSO$, since the map $U(n) \to O(2n)$ factors over $SO(2n)$. But I have not been able to find a reference that gives the example of $MSO$ being complex oriented.

So I guess Q2) is more interesting, already for the simple reason that the ring $MSO_\ast$ is much more delicate than the ring $MO_\ast$. Moreover (as Johannes Ebert also pointed out in a comment in the link above) real oriented theories come equipped with a map from $MO$ and hence split in Eilenberg-MacLane spectra, whereas $MSO$ behaves quite differently.

In addition, I have one last question:

Supposing Q1) and Q2) are true, then as I have already mentioned that $MSO$ is complex oriented. In particular we have a notion of abstract chern classes for $MSO$-cohomology. Moreover we have a canonical map $MSO \to H\mathbb{Z}$. The composite $MU \to H\mathbb{Z}$ is just the usual orientation of integral cohomolgy, in particular the abstract chern classes of this complex oriented theory are just the usual chern classes, but since oriented real bundles have pontryagin classes I ask myself:

Q3) Is there any way to view pontryagin classes as special case of abstract chern classes?

(It seems not to be the case for the cohomology theory $MSO$ which I would have thought of first).