The French company said its sports and entertainment specialists would now be placed "at the core" of Havas Group’s client teams to provide integrated service within the 46 Havas Villages worldwide.
Avery will move on to new projects outside Havas, the company added in an announcement this morning.
One of the most senior Britons in the French agency group, Avery took the role of HSE chief executive in December 2015. He was also chief executive of Arena Media UK and group managing director for Havas Media Group UK until May 2016.
The changes do not affect HSE Cake, Havas' sport and entertainment agency in the UK. HSE Cake's senior management team remains unchanged with Adrian Pettett as chief executive, Jim Dowling as managing director and Joanne Warnes as chief operating officer.
Yannick Bolloré, chairman and chief executive of Havas, said: "My thanks go out to Pedro for his successful management of this transition. It is thanks to his energy and passion that we are able to offer such an incredible expertise.
"Havas Sports & Entertainment is made up of the best experts who truly understand what makes a fan a fan and what drives their engagement.
"Integrating this talent at the heart of all our Havas Villages means all our agencies will be able to deliver solutions and vision for our clients that build emotional relationships between brands, fans and communities based on today's popular culture."
A Havas spokeswoman had not responded to request for more clarification on the move ahead of publication.