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 planets 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.
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