In 1964, the Kickers won the 1964 National Challenge Cup over the Philadelphia Ukrainians.
[1] In 1965, he moved to Orange County Soccer Club of the Continental League.
In 1966, Orange County lost in the final of the 1966 National Challenge Cup.
On January 29, 1967, Abaunza signed with the Los Angeles Toros of the National Professional Soccer League.
The Toros folded at the end of the season and Abaunza moved to the Los Angeles Armenians in 1968 and Los Angeles Saprissa, a team composed almost entirely of Costa Rican players.