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How to compute KO characteristic classes/numbers?

They were introduced by Anderson/Brown/Peterson to study the structure of the spin cobordism ring. I looked through the literature but I did not find a nice example of computation. For instance, I would like to know how to determine $\pi^J(\mathbb{H}P^2)$. Many thanks for any comments in advance.

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Chapter II (in particular sections II.7 and II.9) of the book

Conner, P. E.; Floyd, E. E. The relation of cobordism to K-theories. Lecture Notes in Mathematics, No. 28 Springer-Verlag, Berlin-New York 1966 v+112 pp.

gives a nice construction of $KO$-characteristic classes for symplectic bundles, in particular showing that they satisfy the usual axioms. From this you may be able to perform computations. I'm not sure what the notation $\pi^J$ means (cokernel of the J-homomorphism?).

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