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Galaxy groups with $M_{tot} \lesssim 10^{14}$ $M_\odot$ and up to a few tens of members are the most common galaxy environment, marking the transition between field and massive clusters. Identifying groups plays a crucial role in understanding structure formation and galaxy evolution. Modern deep surveys allow us to build well-characterized samples of groups up to the regime where structures were taking shape. We aimed to build the largest deep catalog of galaxy groups to date over the COSMOS-Web field effective area of 0.45 deg$^2$, leveraging the deep high quality data of the new COSMOS-Web photometric catalog resulted from the James Webb Space Telescope observations of the COSMOS-Web field. We performed the group search with the AMICO algorithm, a linear matched filter based on an analytical model for the group signal. AMICO has already been tested in wide and deep field surveys, including COSMOS data up to $z=2$. In this work, we tested the algorithm performances at even higher redshift and searched for protocluster cores at $z>2$. We compiled a list of known protoclusters in COSMOS at $2 \leq z \leq 3.7$, matched them with our detections and studied the clustering of the detected cores. We estimated purity and completeness of our sample by creating data-driven mocks with the SinFoniA code and linked signal-to-noise to purity. We detected 1678 groups in the COSMOS-Web field up to $z=3.7$, including lists of members extending nearly two magnitudes deeper than the previous AMICO-COSMOS catalog. 756 groups were detected with purity of 80\%. More than 500 groups have their redshift confirmed by assigning spectroscopic counterparts. This group catalog offers a unique opportunity to explore galaxy evolution in different environments spanning $\sim$12 Gyr and to study groups, from the least rich population to the formation of the most massive clusters., Comment: 17 pages, 14 figures, 2 tables (including appendices), submitted to Astronomy & Astrophysics (A&A)