In his new role, Kuniskis will be responsible for the Fiat brand in North America, including sales, marketing, dealer network and service. He will continue the reintroduction of the brand, Chrysler said.
"Tim brings broad expertise and leadership in dealer operations and marketing where he has been already working with the team to shape the direction of the Fiat Brand," Sergio Marchionne, Chrysler CEO and chairman, said in the statement.
"As North American Head of Fiat, much of his immediate focus will be working with the dealer body where his fresh perspective from the operational side, as well as that on the commercial side, will begin the Fiat Brand's next chapter."
Kuniskis joined Chrysler in 1992 and has held a series of increasingly responsible positions in the automaker's business center operations and marketing organization.
Kuniskis, who was head of Chrysler brand product marketing, joined the Fiat brand this summer as Chrysler sought to beef up its marketing and dealership team with a couple veteran Chrysler managers. He oversaw product marketing to help Fiat build its product plan while retaining his Chrysler brand job.
Last week at the Los Angeles Auto Show, Fiat global brand head Olivier Francois conducted the rollout of the North American version of the Fiat 500 Abarth. Soave was at the presentation, but did not participate.
In Kuniskis' previous assignment with the Chrysler brand, he previously worked closely with Francois.
Sales of the Fiat 500 have not kept pace with the automaker's annual target of 50,000 units. The subcompact, which went on sale in March, had sold only 15,826 units (21,380 in North America) through October. Chrysler said it has 130 "customer-ready" Fiat dealerships in the U.S. where customers may order cars, and in October, 29 of them did not record a single sale.
Chrysler Group has suffered a series of setbacks in its rollout of the small subcompact in North America.
Marchionne, during a meeting with journalists in Toledo, Ohio, last week, said the automaker was six months behind in its development of a dealer network.
On Sunday, Jennifer Lopez, the vehicle's new high-profile national spokesperson, had trouble unlocking the door of a Fiat 500 rolled out during a live shot while she was performing on the American Music Awards.
Editor's note: An earlier version of this story misstated the 40-mpg vehicle Fiat expects to use to gain an additional 5 percent stake in Chrysler Group.