The sun is not to blame for almost all of it (just a tiny little bit). This is quite easy to figure out since we have data on the sun and Earth's orbit and the relationship is simple.
It's interesting that the collapse of an ice age seems to happen very fast, within a few hundreds of years, so fast changes are definitely possible.
(It's good to research this since an ice age would suck majorly for us too - and we have suppressed the next one by now.)