  We present a new galaxy cluster search in the COSMOS field through the use of
the Adaptive Matched Identifier of Clustered Objects (AMICO). We produced a new
cluster and group catalogue up to $z=2$, by performing an innovative
application of AMICO with respect to previous successful applications to
wide-field surveys, in terms of depth (down to $r < 26.7$), small area covered
($1.69 deg^2$ of unmasked area) and redshift extent. This sample, and the
comparative analysis we performed with the X-rays, allowed for the calibration
of mass-proxy scaling relations up to $z=2$ and down to less than $10^{13}
M_{sun}$ and constitutes the base for the refinement of the cluster model for
future applications of AMICO, like the analysis of upcoming Euclid data. AMICO
is based on an optimal linear matched filter and detects clusters in
photometric galaxy catalogues using galaxy location, photometric redshift and,
in the simplest case, one galaxy property. We used one magnitude as galaxy
property, avoiding explicit use of galaxy colour, and performed 3 independent
runs in the r, Y and H bands using both COSMOS2020 and COSMOS2015 galaxy
catalogues. The final catalogue resulting from matching the results of the
three runs contains 1269 and 666 candidate clusters with $S/N >3.0$ and $>3.5$,
respectively. Most of the unmatched ones have $S/N <3.5$ which can be chosen as
cut for a more robust sample. We assigned X-ray properties to our detections
via matching with a public X-ray group sample and by estimating, for unmatched
detections, X-ray properties at the location of AMICO candidates based on
Chandra+XMM-Newton data. 622 are the candidates with X-ray flux estimate. This
large sample allowed for the calibration of the scaling relations between AMICO
mass-proxies and X-ray mass and the study of their redshift dependence for the
selection of the most stable photometric bands.
