The PROMOS for SOS, Bang-A-Boomerang, I Do, I Do, I Do, I Do, I Do and Mamma Mia were directed by Lasse Hallström at a combined cost of less than 50,000 Swedish Kronor. These were the four songs that were felt to have the strongest hit potential from the ABBA album and filmed with a view to promote it. 

These four clips were not intended to promote the songs as individual singles but the ABBA album as a whole. The four clips were distributed as one complete film and broadcast on Swedish TV under the name 4 x ABBA.

Mamma Mia was filmed at the same time as I Do, I Do, I Do, I Do, I Do and the studio bit of SOS.

This PROMO film premiered the side-on profile shot of two members which were to go on and be the 'trademark' of ABBA's PROMO videos from then on.

This was the PROMO that started it all in Australia when it was shown on Countdown in August 1975 - see below.

The following is from the ABBA Omnibus blog by Ian Cole:
(with thanks also to Matti Crocker and Trent Nickson)

On Sunday 3 August 1975, Australian TV programme Countdown played the promotional film clip for I Do, I Do, I Do, I Do, I Do. Countdown was a nationally-broadcast weekly music programme on the government-owned ABC network that ran from 1974 to 1987. The show's producers were always in search of popular music to play. They found that a song called I Do, I Do, I Do, I Do, I Do by ABBA was in the top 20 in Brisbane, the capital of the state of Queensland.

Countdown's producers approached RCA, ABBA's Australian record company, requesting any footage they could play. RCA provided four film clips from the group. Countdown was impressed with the clip for the single I Do, I Do, I Do, I Do, I Do, but were even more taken with another song, Mamma Mia.

Mamma Mia was apparently also broadcast at the end of the show on 3 August 1975, in Melbourne (the capital of Victoria) only. But the following Sunday evening, Mamma Mia was broadcast nationally, and something amazing happened. Though the song was not scheduled to be released as a single, public demand on record stores, radio stations, RCA and Countdown led to RCA requesting from Polar Music in Stockholm permission to release Mamma Mia.

Stig Anderson initially refused, claiming that "the Australians had released so many damn singles". In a way he was right: in the 16 months since Waterloo had been released, RCA had put out Ring Ring (its second local release, in remixed form), Honey, Honey, So Long, I've Been Waiting For You (separate singles released simultaneously), and I Do, I Do, I Do, I Do, I Do, plus a four-track EP. After Waterloo's peak at number four in August 1974, ABBA's highest chart position had been Honey, Honey's number 30 peak in January 1975.

Eventually Polar relented, and Mamma Mia entered the singles chart at number 52 on 22 September. In the meantime, I Do, I Do, I Do, I Do, I Do was rushing up the national chart, reaching number one on 13 October. After three weeks at the top, it was replaced by Mamma Mia, which remained at number one for ten weeks. The songs' parent album, ABBA, was caught in the slipstream, entering the album chart on 8 September and reaching number one on 8 December 1975.

The public reaction and quick chart success caught the attention of overseas media. Contemporary UK news programmes reported on the astonishing Australian success of this almost-forgotten Eurovision-winning group, whose latest domestic single SOS was gaining international attention thanks to interest in what was happening in Australia. SOS went on to reach number six in the UK in November, followed by Mamma Mia, which topped the UK chart in February 1976.

Back in Australia, SOS replaced Mamma Mia at number one on 12 January 1976, and a compilation album The Best Of ABBA had been released, topping the chart on 28 March and becoming the first album to sell over one million copies in Australia (ABBA's Arrival later in the year would be the second). 

When ABBA 'returned' on Thursday 2 September 2021, we saw that a six-second out-take from the Mamma Mia PROMO was included in the video for I Still Have Faith In You  (from 1:49).

On Friday 3 February 2023, a new lyric video was posted on YouTube.

ABBA - The Definitive Collection booklet written by Carl Magnus Palm
Thanks to Ian Cole and some info from Bright Lights, Dark Shadows by Carl Magnus Palm
YouTube links c/o AbbaVEVO and ABBA