  Aims: We aim to study the spot evolution and differential rotation in the
magnetically active cool K-type giant star sigma Gem from broadband photometry
and continuous spectroscopic observations that span 150 nights. Methods: We use
high-resolution, high signal-to-noise ratio spectra obtained with the
Hertzsprung SONG telescope to reconstruct surface (photospheric) temperature
maps with Doppler imaging techniques. The 303 observations span 150 nights and
allow for a detailed analysis of the spot evolution and surface differential
rotation. The Doppler imaging results are compared to simultaneous broadband
photometry from the Tennessee State University T3 0.4 m Automated Photometric
Telescope. The activity from the stellar chromosphere, which is higher in the
stellar atmosphere, is also studied using SONG observations of Balmer H alpha
line profiles and correlated with the photospheric activity. Results: The
temperature maps obtained during eight consecutive stellar rotations show
mainly high-latitude or polar spots, with the main spot concentrations above
latitude 45 deg. The spots concentrate around phase 0.25 near the beginning of
our observations and around phase 0.75 towards the end. The photometric
observations confirm a small jump in spot phases that occurred in February
2016. The cross-correlation of the temperature maps reveals rather strong
solar-like differential rotation, giving a relative surface differential
rotation coefficient of $\alpha$ = 0.10 +/- 0.02. There is a weak correlation
between the locations of starspots and enhanced emission in the chromosphere at
some epochs.

Solar and Stellar Astrophysics