I think the secret organization is going to have a few problems here'
First off, define what traits that the "new" humans will have that give them an advantage over H.Sapiens. We are at the top of the food chain precisely because we are smarter, tougher, more social, more dexterous and more adaptable than any other species. Our Ancestors literally walked around the world to settle in habitable places, fought megafauna with pointed sticks and turned objects that other species would ignore into tools. Whatever N.Sapiens has will have to be pretty much off the charts to really make inroads into H.Sapiens
This leads to issue two: strength in numbers. We are sitting at a manpower advantage of 7 billion to 1 for the home team. We cover all the arable land, usable waterways and indeed virtually every piece of the Earth's surface where it is possible to support life. We will probably be able to physically identify N.Sapiens due to physical , mental or emotional features and traits, so can quickly isolate them and physically annihilate them if they are perceived as an existential threat.
Even if they are physically indistinguishable from H.Sapiens (they just keep the prehensile tail tucked into the pants and wear sunglasses 24/7 to hide the Anime eyes...) it is quite possible they cannot breed with normal humans, so their genetic material won't "infiltrate" the human population, and their population will be much smaller and have access to far fewer resources than we do. They will be outbred and outcompeted on virtually every level once you go beyond one on one competition. If they are too specialized, they might not even win there. (For a funny example, think of how Wesley defeated the three kidnappers in "The Princess Bride").
So from a very limited technical perspective, the secret society can indeed create a new type of human being. If they are very patient, they could even do this the "old fashioned" way by selecting people with the desired traits and offering incentives for them to breed with each other, ensuring the desired traits are isolated from the larger gene pool and constantly reinforced. This seems to have happened to the Ashkenazi Jews, for example. The Ashkenazi's also offer a second lesson; be prepared to ruthlessly cull to prevent undesirable traits from also accumulating. To really make the changes widespread you would have to "release" the breeding stock into the main gene pool every few generations, then restart with a "higher" baseline, or you will get a regression to the mean.
Genetic engineering isn't quite there yet (we don't really understand how many genes interact or are expressed), but there is no reason to suppose this won't be figured out within a few decades at most. In that case, the "desirable" traits won't be what the secret society wants, but whatever people are willing to pay for.