I am learning about the AHSS (for complex K-theory) by trying to compute the K-theory of some spaces. I have heard that the AHSS is functorial (maps of spaces induce maps of spectral sequences). Is there a nice example of spaces $X$, $Y$ and a map $X \rightarrow Y$ such that we know the K-theory of one of these spaces and the induced map on their AHSS's helps us to compute the K-theory of the other space? In most of the situations I can think of, either the AHSS of one of the spaces collapses at $E_2$, making the map between the AHSS's useless, or I don't know very much about the AHSS of either space beyond $E_2$, making it hard to begin.
I would also be very interested in where I can find some interesting calculations with the AHSS.