FH picked up the account from 20-month incumbent Band & Brown Communications after a four-way pitch, managed by the Central Office for Information.
The commission wants to foster public confidence in elections, encourage more people to vote, promote transparency and lobby for reform.
The agency will also raise awareness of the commission as the regulator of political parties' financial affairs.
'We want to increase voter engagement and help communicate our role,' said comms director Angela Salt. 'We need to demonstrate that politics is personal. People must be engaged; we need to show them why this matters.'
She added that one challenge will be maintaining public interest in voting between general elections.
FH plans to deliver a PR and stakeholder relations programme. Associate directors Lee Whitehead and Sarah Schofield lead the account, reporting to commission media and PR manager Tabitha Cunniffe.
The commission's recommendations for introducing individual registration, rather than by household, and for tightening up security will be included in a bill due to go through parliament in the autumn.
The campaign will be media relations based, targeting national press, consumer publications such as women's weeklies and glossies, local press, TV and radio chat shows.
Fees are around £100,000 per year.
The commission is currently reviewing its Northern Ireland account (PRWeek, 3 June). Scottish comms are managed by Glasgow-based Barkers Public Relations, with Equinox Public Relations in Wales.
Salt is continuing to recruit for a new head of comms with responsibility for press, corporate comms, public information and 'outreach'.
The commission retains Westminster Strategy for PA and its remit is not affected by FH's appointment.