What is the name of a company's internal IT department?

In most cases the collective name for an IT 'group' within a company is, or could be, called "IT Services" or "Information Services", and both references could be appended with 'Department' if desired.

Nowadays, the Word 'Services' is typically thrown into the mix somewhere as most IT Services 'teams' now, or do their utmost to implement and follow an IT Service Management (ITSM) best practices (notice the Word 'Service'); with [hopefully] most ITSM being drawn up with strong references and methodologies taken from the IT Infrastructure Library (ITIL). ITIL is, as the name puts it, a 'library' for which the underlining governance and guidance that can be [is] used for the 'in-house' ITSM tuning and tweaking to suit the needs of a company's (or customer's) IT service requirements.