[1] Groman attended Heidelberg University and graduated with a bachelor's degree while lettering in three sports that included football.

He taught science for eighth graders in Perrysburg, Ohio, near where he grew up before one day being approached to have lunch with a fellow teacher and her husband Bob Snyder, a former coach in the NFL.

After a round of catch, he realized the talent in Groman and called up a former teammate of his at Notre Dame that happened to have been recently hired for the head role with the Houston Oilers in Lou Rymkus.

In the AFL title game on January 1, 1961, he caught three passes for 37 yards and one touchdown, with his score in the third quarter giving the Oilers a 17–9 lead on the way to a 24–16 win over the Los Angeles Chargers.

Alongside his teammate Charlie Hennigan, they were referred to as "The Long-Distance Twins",[5] with Groman benefitting from his quick feet that made him a track star in college.

During a short catch, he was injured on a tackle by Claude Gibson that resulted in a bent knee from his own description after it collided with a helmet.

Although he did not rack up enough stats to be considered for the Pro Football Hall of Fame, his shoes from the record-breaking rookie season are on display in bronze.