Groves joined Mother in January last year, as its first operations director. Before this he was director of integration at Saatchi & Saatchi, overseeing the acquisition of digital business Outside Line.
He has also held senior roles at Exposure and Fallon.
In his new role Groves will lead Edelman's digital team.
Steve Spurr, chief operating officer, Edelman said: "As digital has matured in our clients' businesses, we needed someone who could help activate our offer across the owned, earned, paid matrix but also advance our 70-strong digital team. Fusing Edelman's earned media mindset with Matt's experience across advertising, design, production and digital will further improve our ability to deliver on the full scope our clients expect from us."