OK, so now I offer a somewhat different answer...
6
For those of us who have used (or still use) the wonderful E-M10 Mk II, there's no denying the fact that this little gem can produce some wonderful IQ levels.
Of course, then all that is left is the lens you wish to attach - based on your budget levels.
I've personally always had this pension for pursuing superb kit lenses - ones that can give the big boys a run for their money without having to go broke. As a result, I've gone through many kit lenses in my quest to find the budget level holy grail.
Now, this is purely based on my opinion - but I've found the Panasonic Lumix 12-60 F3.5-5.6 OIS to be an outstanding addition to the E-M10 Mk II. I have one on my 10 Mk II and has never disappointed. In fact, short of it having a constant F2.8, that lens can give the Leica version of this focal range a great run for the money. In otherwords, this Lumix 12-60 punches far above its weight.
So IMO, keep the E-M10 Mk II and upgrade to the Lumix 12-60. You won't go broke as many are available on eBay - new, separated from the kit (or listed as white box models) and some are going for great prices. I currently have 3 copies of this lens. The first that I purchased was right after it was initially released, I may have been one of the early adopters. I was taken aback at that copy - at how sharp it was, the build quality, and the IQ results. Then I invested in two more - both from eBay and they cost me (new, separated from the kit) at $225 and $279 respectively. And both of those copies were absolutely superb also.
So unless you want to move to a 20 MP model - which there is nothing wrong with that - the E-M10 Mk II is a winner and will be for quite some time. And, it's so called replacement - the Mk III version - has less options than the Mk II did, and many prefer the Mk II version - now go figure.
Currently, here are the Lumix 12-60's available on US eBay (check weekly because more will show up). When you see lots of white box versions - or those separated from kits - it often means that purchasers never gave the kit lens a chance. Rather, they assumed that since it is considered a kit lens, the performance must not be good (and boy is that a false assumption).
US eBay 12-60's:
https://www.ebay.com/sch/i.html?_from=R40&_trksid=p2380057.m570.l1312.R1.TR11.TRC3.A0.H0.XPanasonic.TRS1&_nkw=Panasonic+12-60mm+f%2F3.5-5.6+Lens&_sacat=625
That shows used and new ones.
Several random samples captured with the 12-60 attached to the EM10 Mk II:



