The iFixit Precision Bit Driver is an aluminum or plastic handle that fits 4 mm bits. Found in the Manta, Mako, Moray, Mahi, Minnow, Essential Electronics, Pro Tech and Pro Tech Go Toolkits from iFixit.
If by the "included driver" you are referring to the iFixit Precision Bit Driver that fits 4 mm bits, the quick answer is no. A ¼ (.25) inch (6.35mm) bit will not fit. However, an adapter is available on Amazon although it may not be cost effective. Here's a link: