(Warning, this is a bit long)I had a dog exactly like Lola. She was actually a Cattle Dog mix (which, I think Lola has some of that in her too). I rescued her from New Mexico. I have two small boys, so the idea was to rescue a puppy for them. I have always said that I couldn't understand how people can rehome their pets, because to me once you commit to them, you're taking responsibility for their life- the GOOD and the BAD. Pepper changed that idea for me though, and I guess I'm telling this story so that you might understand why Lola is going through what she is.
I did everything i could with Pepper; obedience, socialization to the MAX, etc. She was super active... I had to run her 3 miles a day and play frisbee at least twice a day just to get her to lie down for a couple hours! She was always "OK" with my kids, but never really cared whether they were around or not. Then, she hit adolescence, and the dominance that I had always seen and worked so hard to keep under control reared it's ugly head and she began to snarl and snap at my boys if they tried to hug her or touch her in the car (I know, there can be issues with kids and provocation, and I assure that there was none. I was always very careful to make sure my boys were always supervised and always treating her with love and respect). I took her to the vet to make sure she wasn't sick, and then to a behavioralist to find out what was going on and how I could rectify it. She told me two things: This Cattle Dog puppy of mine was never going to bond to the boys, because they're a one person breed and I was clearly her person... and that they were specifically bred to be aggressive and dominant so that their little bodies could manage cattle in the outback for 10 hours a day. The second thing she told me is that in Colorado behaviorists see a HUGE number of dogs from New Mexico with people issues. She said no one had an answer as to why, but that all of them share the same anxieties and fears. She told me that while there was a chance that Pepper could be helped, in her opinion, she was more like a time bomb. I already felt I couldn't trust her- and I knew that if she did ever hurt one of my kids, it wouldn't just mean a traumatic situation for my son, but likely she would end up being put down by the city. I felt I had no choice but to find her a home where she could be the dog she was meant to be and remove a potentially dangerous situation from my house. It was the hardest thing I have ever done, and I cried for a week before she left and two weeks after. BUT, now Pepper lives on a Cattle Ranch in Wyoming. She herds cattle all day long until her heart's content. That is the best thing I could have given her! And as much as I miss that little crackling fireball she was, I know that it was the best decision for all of us. If I had known her breed when I rescued her, I would have probably made a different decision, but noone knew what she was or even where she came from!
I just wanted to put my two cents in, because it's often difficult to understand why someone would choose to get rid of an animal (and all too frequently they ARE disposed of as if they're nothing more than property), but sometimes it's for the best. Lola is like Pepper, she needs the right environment to make her shine, and the right type of family to allow her to do that.
Sorry for such a long story.