  We present seventeen high-precision light curves of five transits of the
planet Qatar-2b, obtained from four defocussed 2m-class telescopes. Three of
the transits were observed simultaneously in the SDSS griz passbands using the
seven-beam GROND imager on the MPG/ESO 2.2-m telescope. A fourth was observed
simultaneously in Gunn grz using the CAHA 2.2-m telescope with BUSCA, and in r
using the Cassini 1.52-m telescope. Every light curve shows small anomalies due
to the passage of the planetary shadow over a cool spot on the surface of the
host star. We fit the light curves with the prism+gemc model to obtain the
photometric parameters of the system and the position, size and contrast of
each spot. We use these photometric parameters and published spectroscopic
measurements to obtain the physical properties of the system to high precision,
finding a larger radius and lower density for both star and planet than
previously thought. By tracking the change in position of one starspot between
two transit observations we measure the orbital obliquity of Qatar-2 b to be
4.3 \pm 4.5 degree, strongly indicating an alignment of the stellar spin with
the orbit of the planet. We calculate the rotation period and velocity of the
cool host star to be 11.4 \pm 0.5 d and 3.28 \pm 0.13 km/s at a colatitude of
74 degree. We assemble the planet's transmission spectrum over the 386-976 nm
wavelength range and search for variations of the measured radius of Qatar-2 b
as a function of wavelength. Our analysis highlights a possible H2/He Rayleigh
scattering in the blue.
