Open to pupils of all faiths and ages, it creates a Jewish community in a non-Jewish environment, and is an integral part of a school's informal education package.
"This is one day a week when we can hear about Judaism," says Josh, the society's co-chair, who also happens to be Head Boy. "Non-Jews and Jews are welcome - for some, it's their only chance to learn something about it."
It is two years since UJIA took the helm of JAMS, which it runs alongside its continued activities inside Jewish schools.
Since then, the programme has reached 1,200 students and now caters for 23 London-based schools, plus a further eight in Manchester and Glasgow. As well as funding the running costs of each school's JSoc, it provides the food, speakers and a term calendar of key events. The aim is to equip students in non-Jewish schools with the tools to strengthen their Jewish identities and connections to Israel.
"We have an extensive brochure full of speakers that showcase a whole range of the community," explains Debra Green, UJIA's schools' informal education manager, who oversees the programme. "We give each school a budget and a list of over 150 organisations and individuals to run sessions, and then they send their booking forms to me. It teaches them to take initiative, which prepares them for life after school.
"By learning these social and financial skills, the programme develops peer leadership. We're creating our future leaders."
Each JSoc is tailored to suit the needs and demands of their particular institution. While some have a large and thriving Jewish student body, and therefore meet more than once a week, others gather their lesser numbers once or twice a term, offering alternatives to the school's choral concerts or Christian assemblies.
Sessions range from celebrating Jewish festivals and events to those covering Jewish history, religious values, social action, health and wellbeing.
They also devote at least 20 per cent of their time to Israel, giving insight into a country that many know nothing about, not least the non-Jewish students who come along. Last year, 691 UJIA JAMS sessions took place across the country.
Students who then wish to take their involvement with UJIA further may sign up for UR UNI LIFE, a programme run with the Union of Jewish Students (UJS) that prepares them for life on a university campus after leaving school.
At Watford Grammar, more than 20 pupils get together once a week. Over a hot lunch - on this particular occasion, kosher meatballs and rice - they engage in philosophical debates and ponder religion's bigger questions.
"My Barmitzvah is in September - this is giving me a good way to prepare for it," says 12-year-old Andrew Hodes. "My favourite thing is the food, but I enjoy the talks too."
17-year-old Niall Sharma reveals he came along for the first time to find out more about Judaism. "I'm Hindu, but I like learning about other cultures," he says. "This gives me a bit more information."
On this particular occasion, Rabbi Ari Faust, shaliach for the Jewish Agency to Bnei Akiva, is leading the session questioning God. For him, the UJIA JAMS sessions are "incredible".
"I have so far visited four or five schools and have seen so much interest, both from Jews and non-Jews," he says.
"I myself grew up in a strictly orthodox environment. It is great to see children in a secular school taking control of their Jewish identity."
And while the majority of UJIA JAMS programmes are London-based, their influence is often considered more beneficial outside the capital, where there are very few Jewish secondary schools.
This means that many pupils who experience a Jewish education until the age of 11 can enter a secular secondary school feeling disconnected from their community, especially if they are not involved in any youth movements. This is where UJIA JAMS steps in.
"Sometimes the Jewish assembly is the cool thing to go to," reveals Julia Rae-King, who, as a UJIA JAMS youth worker, oversees their Glasgow programmes. "The Jewish pupils bring along their pals, who get to learn something new.
"We're in four schools in total. It gives kids in secondary schools the chance to continue their Jewish learning and explore their Jewish identity. I think it's really important to offer these sessions so that they don't fall through the cracks and just end up going along with the Christian majority."
She adds that "they particularly love the smoked salmon".
