We present 17 high-precision light curves of five transits of the planet Qatar-2 b, obtained from four defocused 2 m-class telescopes. Three of the transits were observed simultaneously in the Sloan g ′ r ′ i ′ z ′ passbands using the seven-beam Gamma Ray Burst Optical and Near-Infrared Detector imager on the MPG/ESO 2.2-m telescope. A fourth was observed simultaneously in Gunn grz using the Centro Astronómico Hispano Alemán 2.2-m telescope with Bonn University Simultaneous Camera, 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 star-spot between two transit observations, we measure the orbital obliquity of Qatar-2 b to be λ = 4 $_{.}^{\circ}$ 3 ± 4 $_{.}^{\circ}$ 5, 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 11.5 ± 0.2 d and 3.28 ± 0.04 km s−1 at a colatitude of 74°. 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.