The FHA does permit streamline refinances on insured mortgages. The term "streamline" refers to the amount of documentation and underwriting which needs to be performed by the lender, and does not mean there are no costs involved in the transaction.
The Department of Housing and Urban Development (HUD) was created in 1965. HUD over sees the Federal Housing Administration, recognized commonly as FHA.
The FHA's duty is to administer government loans and to set the FHA loan requirements for the government loan guarantee program.