The Watch the Med association said it received a call for help from the stricken inflatable at around 0830 GMT on Wednesday.
"The authorities were unreachable," Watch the Med said of the Libyan coastguard charged with search and rescue operations in the zone off Zouara.
Charities frequently accuse the Libyan authorities of failing to rescue migrants, while rights groups say migrants face abuses in Libya.
Instead, German charity Sea-Watch's Alan Kurdi rescue vessel, named after a Syrian boy whose body washed up on a Turkish beach in 2015, went to rescue the migrants.
"They're all safe and sound on board our vessel," Sea-Eye said.
🇪🇺 Europa rettet noch. Die Crew der #AlanKurdi hat heute 64 Menschenleben bewahrt, doch noch werden 50 Menschen vermisst. Die Libyer kommunizieren und retten nicht. Wir tun das, mit eurer Hilfe, denn #unserEuroparettet auf https://t.co/K9VBQ4Q6TX #saveLives #MenschenstattGrenzen pic.twitter.com/JXd7BrcsGM
— sea-eye (@seaeyeorg) April 3, 2019
"The Libyan coastguard is not answering or rescuing."
Italy's far right Interior Minister Matteo Salvini made his oft-stated hardline position on immigration crystal clear Wednesday evening.
"A German-registered vessel, a German NGO, a German ship owner and a skipper from Hamburg — best it heads for Hamburg," said Salvini.
Italy's tough line on the issue has seen many boats that pick up migrants making the perilous journey across the Mediterranean increasingly return them to chaos-wracked Libya.
But there they face trafficking, kidnap, torture and rape, according to the United Nations and aid groups.
Watch the Med said it had received distress calls from two other vessels which had left Libya in recent days with a total 91 people aboard.
The Italian coastguard said Tuesday they had information on the second of the two and had passed on the information to their Libyan counterparts.
But according to the International Organisation Migration (IOM) there was "still no news" of either, despite the Alan Kurdi vessel spending several days searching the zone.
"Sadly, that confirms the search and rescue capacities in the Mediterranean have dropped and must be ratcheted up," OIM spokesman Flavio Di Giacomo said.
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