often they do, especially the bigger companies that can afford it.
Depends on the company and type of work involved. if they use an outside company.. they usually check 75%. In house maybe 50%
Depends on the employer, HR and/or manager thinking of hiring you. If you lie, it will come out, maybe not now, maybe not for a while, but eventually, it will be exposed.
If they care about quality hires and are connected in the industry, they might do at least some "social research" (Googling you, scanning your social media profiles, asking around their network if there's a connection to you, etc.) Don't lie. It will come back against you! Red flags are the quickest way to get ignored.
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