Hi Anon,
There are many government units that offer jobs for IT Engineers but unfortunately the number of jobs in itself is very low. Almost every public sector undertaking requires IT Professionals, but where such units require thousands of LDC, hundreds of UDC & Technicians per establishment they need only a few IT Engineers. Almost all IT work is outsourced, role of IT engineer in a government unit is not to built systems but to run them and provide consultation to IAS on the floor above you (literally) .
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Now coming to your question, There are many Central and public sector undertakings, banking units, semi government units that require IT Services.
PSU : You get recruited to PSUs either by GATE, or by applying directly on their call but for such positions you must have an experience of 5–10 years & certifications like CCNA, CCNP, RHCE, OCP , SAP, or C-DAC. If you want to target PSUs then I would recommend going through GATE scorecard only, for two reasons