  Context. Centaurs go around the Sun between the orbits of Jupiter and
Neptune. Only a fraction of the known centaurs have been found to display
comet-like features. Comet 29P/Schwassmann-Wachmann 1 is the most remarkable
active centaur. It orbits the Sun just beyond Jupiter in a nearly circular
path. Only a handful of known objects follow similar trajectories.
  Aims. We present photometric observations of 2020 MK4, a recently found
centaur with an orbit not too different from that of 29P, and we perform a
preliminary exploration of its dynamical evolution.
  Methods. We analyzed broadband Cousins R and Sloan g', r', and i' images of
2020 MK4 acquired with the Jacobus Kapteyn Telescope and the IAC80 telescope to
search for cometary-like activity, and to derive its surface colors and size.
Its orbital evolution was studied using direct N-body simulations.
  Results. Centaur 2020 MK4 is neutral-gray in color and has a faint, compact
cometary-like coma. The values of its color indexes, (g'-r')=0.42+/-0.04 and
(r'-i')=0.17+/-0.04, are similar to the solar ones. A lower limit for the
absolute magnitude of the nucleus is Hg=11.30+/-0.03 mag which, for an albedo
in the range of 0.1-0.04, gives an upper limit for its size in the interval
(23, 37) km. Its orbital evolution is very chaotic and 2020 MK4 may be ejected
from the Solar System during the next 200 kyr. Comet 29P experienced relatively
close flybys with 2020 MK4 in the past, sometimes when they were temporary
Jovian satellites.
  Conclusions. We confirm the presence of a coma of material around a central
nucleus. Its surface colors place this centaur among the most extreme members
of the gray group. Although its past, present, and future dynamical evolution
resembles that of 29P, more data are required to confirm or reject a possible
connection between the two objects and perhaps others.
