It was part of the Wirral Rural District from 1894 until 1933. In 1933 Wirral Rural District was abolished and Caldy became part of the new Hoylake Urban District. [There is conflicting evidence here. Both GENUKI and A Vision of Britain through Time state that Caldy was transferred to Hoylake Urban District in 1933, but Wikipedia says Wirral Urban District.]
The population was 90 in 1801, 138 in 1851, 248 in 1901, and 413 in 1951. At the time of the 2001 Census, Frankby had 310 inhabitants.
In the nationwide municipal reorganization of 1974 the entire Wirral Peninsula, with the exception of the municipal borough of Ellesmere Port and urban district of Neston at the southern end, became the Metropolitan Borough of the Wirral in the new administrative county of Merseyside.
For more information, see the EN Wikipedia article Frankby.